Better-Quasi-Order: Ideals and Spaces

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Better-quasi-order: ideals and spaces

Ainsi de suite…

Yann Pequignot July 2015

Last update: 2nd March 2016 Université Paris Diderot – Paris 7 Université de Lausanne

Ph.D. Thesis

Better-quasi-order: ideals and spaces

Yann Pequignot

The 23rd of July 2015

Supervisors: Jacques Duparc Jean-Éric Pin

Referees: Claude Laflamme Maurice Pouzet

Jury: Alessandro Andretta Valérie Chavez Alain Louveau 3 To Lolita Bien que les pieds de l’homme n’occupent qu’un petit coin de la terre, c’est par tout l’espace qu’il n’occupe pas que l’homme peut marcher sur la terre immense. Bien que l’intelligence de l’homme ne pénètre qu’une parcelle de la vérité totale, c’est par ce qu’elle ne pénètre pas que l’homme peut comprendre ce qu’est le ciel.

— Tchouang-tseu

5 Acknowledgements

I would first like to thank Jacques Duparc and Jean-Éric Pin, mytwo supervisors. They have placed their trust in me and have always supported me wherever my ideas have led me.

I acknowledge the support of the Swiss National Science Foundation and the University of Lausanne. I consider myself very fortunate to have had the chance to carry out this work under optimal conditions.

Part of this thesis is the result of a collaboration with Raphaël Carroy who believed in me when I needed it. I thank him deeply for his friendship and continuous support.

Having shared our interest in the foundations of mathematics from the beginning, Kevin Fournier has been both a dear friend and a very reliable colleague all these years. His company means a lot to me.

I am very grateful to the descriptive set theory team of Paris 6, and specifically to Gabriel Debs, Dominique Lecomte and Alain Louveau. They not only gave me the opportunity to present most of the material in this thesis on the blackboard, but they also kindly insisted on me explaining and justifying every detail. Their advice and demanding nature contributed greatly in the process of completing my thesis.

For his support and advice, as well as for his very kind hospitality and warm welcome in Lyon, I thank Maurice Pouzet.

The pursuit of my thesis was not without several moments of serious doubt. For their encouragement in those difficult times, I gratefully thank Serge Grigorieff and Mai Gehrke.

The path towards the completion of this thesis was sinuous, to say the least. I am indebted to many other people for contributing to my journey, each in their own unique way. Amongst them are Claude Laflamme, Georges Hansoul, Stevo Todorčević, Jaroslaw Nešetřil, Alberto Marcone, Alessandro

6 Andretta, Olivier Finkel, Sławomir Solecki, Boban Veličković, Mirna Džamonja, Norbert Sauer, Arno Pauly, Vasco Brattka, Ralf Schindler, Gerhard Jäger and Thomas Strahm, Julien Melleray, Andreas Blass, Lionel Nguyen van Thé.

I also want to thank the anonymous reviewers of my papers. With their expertise and very careful reading, they provided me with some extremely valuable feedback.

While pursuing my thesis I had the chance to travel and meet many very interesting young researchers. For stimulating discussions as well as priceless moments of camaraderie, I thank Sam van Gool, Gregory McKay, Denis Küperberg, Charles Papperman, Luc Dartois, Rafael Zamora, Laure Daviaud, David Chodunský, Claribet Piña, Honza Starý, Daisuke Ikegami, Omar Selim, Barnabás Farkas, Alessandro Facchini, Zoltán Vidnyánszky, Rukiye Cavus, Dandolo Flumini, Marion Hämmerli, Kentaro Sato, Dimitrios Vlitas, Clinton Conley, Thilo Weinert.

I thank my family for their love and caring support. I am especially thinking of my mother who, in the first place, gave me the passion to seek understanding and who always had confidence in my ability to find my answers.

A number of incredible people have accompanied me through this part of my life by giving me their support, friendship and love. Lucas, Jeremy, Rafael, Deborah, Alexandre, Bruno, Wilhelm, Noémie, Louis, Luc, Delphine, Mathias, Kevin, Raphaël, François, Gregory, Marc, Alessandro, Laure, Jérémie and, especially, Sarah.

There are more. I thank you all wholeheartedly.

Lausanne, March 2015, Yann Pequignot

7 Contents

1 Introduction 1 1.1 From well to better ...... 3 1.2 A well-quasi-order on the subsets of a topological space .... 6 1.3 Organisation of the thesis ...... 10

2 Sequences in sets and orders 15 2.1 Well-quasi-orders ...... 15 2.1.1 Good versus bad sequences ...... 16 2.1.2 Subsets and downsets ...... 18 2.1.3 Regular sequences ...... 20 2.1.4 Sequences of subsets ...... 21 2.2 Better-quasi-orders ...... 24 2.2.1 Towards better ...... 24 2.2.2 Super-sequences ...... 28 2.2.3 Multi-sequences ...... 35 2.2.4 Iterated powerset, determinacy of finite games ..... 37 2.2.5 Equivalence ...... 43 2.3 Around the definition of better-quasi-order ...... 45 2.3.1 The perfect versus bad dichotomy ...... 46 2.3.2 Generalised shifts ...... 48

3 Sequences in spaces 52 3.1 Cauchy super-sequences ...... 53 3.2 Uniform subspaces of Boolean spaces ...... 58 3.3 Cauchy super-sequences ...... 61 3.4 Converging super-sequences ...... 66 3.4.1 Normal form ...... 66 3.4.2 Super-sequences of isolated points ...... 69

4 The ideal space of a well-quasi-order 72 4.1 Completions of well-quasi-orders ...... 74 4.1.1 Ideal completion ...... 74 4.1.2 The Cauchy ideal completion ...... 76

8 Contents

4.1.3 Duality and Cauchy ideal completion ...... 81 4.1.4 Profinite completion, Nachbin order-compactification . 87 4.2 From well to better: A proof of Pouzet’s conjecture ...... 93 4.3 Corollaries ...... 97 4.3.1 Finitely many non principal ideals ...... 97 4.3.2 Interval orders ...... 98 4.3.3 Classes of better-quasi-orders via forbidden patterns . 100

5 A Wadge hierarchy for second countable spaces 102 5.1 The van Engelen–Miller–Steel theorem ...... 102 5.2 Reductions as total relations ...... 113 5.3 Relatively continuous relations ...... 114 5.3.1 Admissible representations ...... 115 5.3.2 Relative continuity ...... 119 5.3.3 Admissible representations and dimension ...... 120 5.3.4 Relatively continuous relations ...... 122 5.3.5 Reduction by relatively continuous relations ...... 125 5.4 Borel and Hausdorff–Kuratowski hierarchies ...... 126 5.5 A general reduction game ...... 132 5.6 The real line and the Scott domain ...... 136 5.6.1 The Real Line ...... 136 5.6.2 The Scott Domain ...... 139

6 Conclusion 142

List of Symbols 145

Index 147

Bibliography 149

9 List of Figures

2.1 Rado’s poset ℜ...... 25 2.2 Pictures of fronts ...... 29 2.3 Constructing a multi-sequence by stringing strategies together. 40 2.4 Stringing strategies together...... 43 2.5 Copying and shift...... 45

∗ 3.1 A front 퐹 with rk 퐹 = 휔 + 1 and |퐹 |CB = 휔...... 56 3.2 A block of the front [휔]2...... 61 3.3 A front whose derivative of the closure is no closure of a front. 70

4.1 Examples of partial orders without property 푀...... 79 4.2 Hamburger diagram: left vs right, algebra vs topology. .... 84 4.3 A bad Cauchy super-sequence into Rado’s poset...... 95

5.1 A play of the game 퐺(푙0, 푙1)...... 105 5.2 Stringing strategies together towards infinity...... 107 5.3 Consequences of Wadge Lemma ...... 112 5.4 Relatively continuous relation...... 122 5.5 The Borel hierarchy ...... 127

10 1 Introduction

Mathematicians have imagined a myriad of objects, most of them infinite, and inevitably followed by an infinite suite. What does it mean to understand them? How does a mathematician venture to make sense of these infinities he has imagined? Perhaps, one attempt could be to organise them, to arrange them, to order them. At first, the mathematician can try to achieve this in a relative sense by comparing the objects according to some idea of complexity; this object should be above that other one, those two should be side by side, etc. So the graph theorist may consider the minor relation between graphs, the recursion theorist may study the Turing reducibility between sets of natural numbers, the descriptive set theorist can observe subsets of the Baire space through the lens of the Wadge reducibility or equivalence relations through the prism of the Borel reducibility, or the set theorist can organise ultrafilters according to the Rudin-Keisler ordering. This act of organising objects amounts to considering an instance of the very general mathematical notion of a quasi-order (qo), namely a transitive and reflexive relation. As a means of classifying a family of objects, the following property of a quasi-order is usually desired: a quasi-order is said to be well-founded if every non-empty sub-family of objects admits a minimal element. This means that there are minimal – or simplest – objects which we can display on a first bookshelf, and then, amongst the remaining objects there are again simplest objects which we can display on a second bookshelf above the previous one, and so on and so forth – most probably into the transfinite. However, as a matter of fact another concept has been ‘frequently discovered’ [Kru72] and proved even more relevant in diverse contexts: a well-quasi-order (wqo) is a well-founded quasi-order which contains no infinite antichain. In- tuitively a well-quasi-order provides a satisfactory notion of hierarchy: as a well-founded quasi-order, it comes naturally equipped with an ordinal rank and there are up to equivalence only finitely many elements of any given rank. To prolong our metaphor, this means that, in addition, every bookshelf displays only finitely many objects – up to equivalence. The theory of wqos consists essentially of developing tools in order to show

1 1 Introduction that certain quasi-orders suspected to be wqo are indeed so. This theory exhibits a curious and interesting phenomenon: to prove that a certain quasi-order is wqo, it may very well be easier to show that it enjoys a much stronger property. This observation may be seen as a motivation for considering the complicated but ingenious concept of better-quasi-order (bqo) invented by Crispin St. J. A. Nash-Williams [Nas65]. The concept of bqo is weaker than that of well-ordered set but it is stronger than that of wqo. In a sense, wqo is defined by a single ‘condition’, while uncountably many ‘conditions’ are necessary to characterise bqo. Still, as Joseph B. Kruskal [Kru72, p.302] observed in 1972: ‘all “naturally occurring” wqo sets which are known are bqo’1. The first contribution of this thesis is to the theory of wqo and bqo. The main result is the proof of a conjecture made by Maurice Pouzet [Pou78] which states that any wqo whose ideal completion remainder is bqo is actually bqo. Our proof relies on new results with both a combinatorial and a topological flavour concerning maps from a front into a compact metric space. We think that these results are of independent interest and hope that they can be applied in other situations where fronts and barriers are used, as in the theory of Banach spaces for example. Our second contribution is of a more applied nature and deals with topological spaces. We define a quasi-order on the subsets of every second countable 푇0 topological space in a way that generalises the Wadge quasi-order on the Baire space, while extending its nice properties to virtually all these topological spaces. Our starting point is the celebrated Wadge quasi-order – of reducibility by continuous functions – on subsets of the Baire space. This quasi-order is described by Alessandro Andretta and Alain Louveau [AL] as ‘the ultimate analysis of the subsets of the Baire space’. The fact that the extremely fine Wadge quasi-order is wqo on Borel sets is doubtless among the most attractive of its properties. The proof of this fact given by Tony Martin, building on previous work by Leonard Monk, is an example of one of the main techniques of bqo theory, namely the use of infinite games and determinacy. Moreover, aswe explain in this thesis, this property of the Wadge quasi-order follows from an extension of the idea underlying the very definition of a bqo. For other important topological spaces the quasi-order of reducibility by continuous functions is however far less satisfactory. For example, the family of Borel subsets of the real line is very far from being wqo under continuous reducibility. While reducibility by discontinuous functions has been studied by

1The minor relations on finite graphs, proved to be wqo by Robertson and Seymour [RS04], is to our knowledge the only naturally occurring wqo which is not yet known to be bqo.

2 1 Introduction some authors to remedy this situation, we propose instead to keep continuity but to weaken the notion of function to that of relation. Using the notion of admissible representation studied in Type-2 theory of effectivity, we define the quasi-order of reducibility by relatively continuous relations. We show that this quasi-order both refines the classical hierarchies of complexity andis wqo on the Borel subsets of virtually every second countable 푇0 space.

1.1 From well to better

A quasi-order 푄 is a wqo if it contains no infinite descending chain nor infinite antichain. However using Ramsey’s Theorem this is equivalent to the absence of a so-called bad sequence, namely a sequence (푞푛)푛∈휔 such that 푚 < 푛 in 휔 ⩽̸ implies 푞푚 푞푛 in 푄. The concept of better-quasi-order was invented by Nash-Williams [Nas65]. Its definition relies on a generalisation of Ramsey’s Theorem to transfinite dimension: the notion of front. It generalises the definition of wqo given above in the sense that it does not only forbid bad sequences, but also bad sequences of sequences, bad sequences of sequences of sequences and so on and so forth in the transfinite. A front can be thought of as a convenient notion ofindex sets for these sequences of … of sequences and we call any map from a front into some set a super-sequence.A bqo is then a quasi-order which admits no bad super-sequence. One contribution of this thesis is to show that super-sequences deserve their name since they share significant properties with usual sequences. A crucial property for a sequence in the context of metric spaces is the Cauchy condition. In order to generalise the notion of being Cauchy to super-sequences, we observe that a sequence (푥푛)푛∈휔 in a metric space 풳 satisfies the Cauchy condition if and only if the mapping 휔 → 풳, 푛 ↦ 푥푛 is uniformly continuous, when 휔 is identified with a subspace of the Cantor space 2휔 via 푛 ↦ 0푛10휔. As observed notably by Todorčević [AT05; Tod10], fronts can naturally be seen as subsets of the Cantor space. Being a compact Hausdorff space, the Cantor space admits a unique uniformity that is compatible with its topology. Even though a front is a discrete topological subspace of 2휔, we observe that it inherits a non-trivial uniformity from 2휔. Let us say that a super-sequence in a metric space is Cauchy when it is uniformly continuous. We show the following theorem, which generalises the usual sequential compactness for metric spaces.

Theorem 1.1 (with R. Carroy). Every super-sequence in a compact metric space has a Cauchy sub-super-sequence.

3 1 Introduction

This combinatorial result should be compared with Erdös-Rado Theorem [ER50] and Pudlak-Rödl Theorem [PR82] as a Ramsey theorem for partitions into infinitely many classes. We also note also that this result subsumes Nash- Williams’ Theorem. Given a complete metric space 풳, every Cauchy sequence 푓 ∶ 휔 → 풳 converges, and thus extends to a continuous map 푓 ∶ 휔 → 풳, where 휔 is the one point compactification of 휔. The same is true about Cauchy super-sequences: any uniformly continuous super-sequence 푓 ∶ 퐹 → 풳 from a front 퐹 into a complete metric space 풳 continuously extends to the uniform completion 퐹 of 퐹 , which coincides with the topological closure of the front inside the Cantor space, to yield a continuous map 푓 ∶ 퐹 → 푋. We also study the continuous extension of Cauchy super-sequences. In full generality, we are concerned with continuous maps from the topological closure of a front into some topological space. Recall that a point 푥 in a topological space 풳 is called isolated if the singleton {푥} is open in 풳, and limit otherwise. The following simple fact exhibits a property of converging sequences that can always be achieved by going to a subsequence: If (푥푛)푛∈휔 is a sequence converging in a topological space 풳 to some point 푥, then there is a subsequence (푥푛)푛∈푁 such that

1. if 푥 is isolated, then (푥푛)푛∈푁 is constant equal to 푥; 2. if 푥 is limit, then

either 푥푛 is isolated for all 푛 ∈ 푁;

or 푥푛 is limit for all 푛 ∈ 푁. We generalise this fact to super-sequences by notably showing the following result.

Theorem 1.2 (with R. Carroy). Let 푓 ∶ 퐹 → 풳 be a continuous extension of a super-sequence 푓 in a topological space 풳. Then there exists a sub-super- sequence 푓′ ∶ 퐹 ′ → 풳 of 푓 such that either 푓′ ∶ 퐹 ′ → 푋 is constant and equal to an isolated point; or {푠 ∈ 퐹 ′ ∣ 푓(푠) is limit} = 퐺 for some front 퐺.

We then apply these theorems to the theory of bqo. The main result is a proof of a conjecture made by Pouzet [Pou78] in his thèse d’état. By an ideal of a quasi-order 푄 we mean a downward closed and up-directed subset of 푄. To every element 푞 ∈ 푄 corresponds the principal ideal ↓ 푞 = {푝 ∈ 푄 ∣ 푝 ⩽ 푞}.

4 1 Introduction

The ideal completion of 푄 is defined as the set of ideals of 푄 partially ordered by inclusion, and it is denoted by Id(푄). Notice that 푄 embeds into Id(푄) via the map 푒 ∶ 푞 ↦ ↓ 푞. We denote by Id*(푄) the set Id(푄)∖푒(푄) of non-principal ideals of 푄, this is the remainder of the ideal completion of 푄. The statement of the conjecture made by Pouzet [Pou78] is the following:

Theorem 1.3 (with R. Carroy). If 푄 is wqo and Id*(푄) is bqo, then 푄 is bqo.

Pouzet and Sauer [PS06] advanced a proof of this statement, but their proof contains a gap, as clearly revealed by Alberto Marcone and acknowledged by Pouzet and Sauer. While the approach of Pouzet and Sauer [PS06] is purely combinatorial, we follow a completely different line and make essential use of the fact that the ideal completion of wqo admits a natural compact topology. We view the importance of the ideal completion as the coincidence in the case of a wqo of several notions of completions of a quasi-order. In fact, gathering many results and facts which certainly belong to the folklore we obtain the following:

Theorem 1.4. For a wqo 푃 the following completions coincide:

(i) the ideal completion Id(푃 ) equipped with the Lawson topology,

(ii) the Cauchy ideal completion of 푃 ,

(iii) the Nachbin order-compactification, or ordered Stone-Čech compactification, of 푃 with the discrete topology.

The different properties of these three different completions combine togive what we call the ideal space of a wqo. This enables us to show that Theorem 1.1 admits the following nice corollary in the context of wqo theory.

Theorem 1.5 (with R. Carroy). Every super-sequence 푓 ∶ 퐹 → 푄 into a wqo 푄 admits a Cauchy sub-super-sequence 푓′ ∶ 퐹 ′ → 푄, which therefore extends to a continuous map 푓′ ∶ 퐹 ′ → Id(푄) into the ideal space of 푄.

As a matter of fact, the ideal space of a wqo is a scattered compact space whose limit points are exactly the non principal ideals. Applying Theorem 1.2, this allows us to prove that any bad super-sequence in a wqo 푄 yields a bad super-sequence into the non principal ideals of 푄. Therefore proving Pouzet’s conjecture.

5 1 Introduction

1.2 A well-quasi-order on the subsets of a topological space

The versatile concept of a topological space has proved valuable in various areas of mathematics. In many cases of interest, the spaces are second countable, i.e. their topology admits a countable base. While separable metrisable spaces are of primary importance to Analysis [Kec95], topological spaces that do not satisfy the Hausdorff separation property are central to Algebraic Geometry [EH00] and to Computer Science [Gou13]. We consider without distinction all second countable spaces which satisfy the weakest separation property 푇0, namely every two points which have exactly the same neighbourhoods are equal. We are interested in finding a way to quasi-order the subsets of a topological space according to their complexity. Among the properties of such a quasi- ordering the following are arguably desired.

• It should agree with an a priori idea of topological complexity, in particular it should refine the classical hierarchies.

• It should be as fine as possible.

• It should be wqo or even bqo – at least on Borel subsets.

The very act of defining a topology on a set of objects consists in specifying simple, easily observable properties: the open sets. We are then interested in understanding the complexity of the other subsets relatively to the open sets. Already at the turn of the twentieth century, the French analysts – Baire, Borel and Lebesgue – stratified the Borel sets of a metric space into a transfinite 0 0 0 hierarchy: the Baire classes 횺훼, 횷훼 and 횫훼. These classes are well-known to exhibit the following pattern:

횺0 횺0 횺0 1 ⊆ 2 ⊆ ⊆ ⊆ 훼 ⊆ ⊆ 횫0 ⊆ 횫0 ⋯ 횫0 ⊆ 횫0 ⋯ 2 ⊆ 3 훼 ⊆ 훼+1 0 ⊆ 0 ⊆ ⊆ ⊆ 0 ⊆ ⊆ 횷1 횷2 횷훼

Borel sets are thus classified according to the complexity of their definition from open sets along this transfinite ladder. This classification was further refined by Hausdorff, and later by Kuratowski, by identifying what isnow called the difference hierarchies, consisting of the Hausdorff–Kuratowski classes 0 −1 퐷휉(횺훼). Since for every map 푓 ∶ 풳 → 풳, the preimage function 푓 ∶ 풫(풳) →

6 1 Introduction

풫(풳) is a complete Boolean homomorphism, it directly follows from their definition that the Borel classes and the Hausdorff–Kuratowski classes are closed under continuous preimages2. Wadge in his Ph.D. thesis [Wad82] was the first to investigate the quasi-order of continuous reducibility on the subsets of the Baire space 휔휔: for 퐴, 퐵 ⊆ 휔 휔 we say that 퐴 is Wadge reducible to 퐵, 퐴 ⩽W 퐵, if and only if there exists a continuous 푓 ∶ 휔휔 → 휔휔 such that 푓−1(퐵) = 퐴. This quasi-order called the Wadge quasi-order relates to the complexity of the subsets of the Baire space in the sense that 퐴 ⩽W 퐵 if and only if one can continuously reduce the membership problem for 퐴 to the membership problem for 퐵. This quasi-order is remarkable. By considering suitable infinite games and using the determinacy of these games, which follows from Borel determinacy, this quasi- order turns out to be well-founded and to admit antichains of size at most 2 on the Borel sets. As Andretta and Louveau [AL] describe in their introduction to [KLS12]: ‘The Wadge hierarchy is the ultimate analysis of 풫(휔휔) in terms of topological complexity’. While the Borel classes and the Hausdorff–Kuratowski classes are closed under continuous preimages, and therefore represent initial segments for ⩽W, there are in fact many more initial segments, so that the Wadge qo refines greatly these classical hierarchies. Of course the quasi-order of continuous reducibility can be defined in any topological space 풳 in the obvious way, for 퐴, 퐵 ⊆ 풳 let 퐴 ⩽W 퐵 if and only if there exists a continuous function 푓 ∶ 풳 → 풳 such that 퐴 = 푓−1(퐵). The nice properties of the Wadge quasi-order extend easily to all zero-dimensional Polish spaces, or even to all Luzin – or Borel absolute – zero-dimensional spaces. The restriction to Luzin spaces and their Borel subsets comes from the use of determinacy in the proof, but it can be weaken if one is willing to assume the determinacy of a larger class of games. In particular assuming the Axiom of Determinacy, the same holds for the quasi-order of continuous reducibility on all subsets of any zero-dimensional second countable space. However the restriction to zero-dimensional spaces is of a different nature. In fact when the space is not zero-dimensional there may be very few continuous functions, independently of any determinacy hypothesis. Hertling in his Ph.D. thesis [Her96] shows that the qo of continuous reducibility of the Borel subsets of the real line ℝ exhibits a more complicated pattern than in the case of the Baire space. For example, Ikegami showed in his Ph.D. thesis [Ike10] (see also [IST]) that the powerset 풫(휔) of 휔 partially ordered by inclusion modulo finite – and hence any partial order of size ℵ1 – embeds in the qo of continuous

2i.e. for every 퐴 ⊆ 풳 in the class and every continuous 푓 ∶ 풳 → 풳 the set 푓−1(퐴) belongs to the class.

7 1 Introduction reducibility of Borel sets of the real line (cf. Subsection 5.6.1). In a more general setting, Schlicht showed in [Sch] that in any non zero-dimensional metric space there is an antichain for the qo of continuous reducibility of size continuum consisting of Borel sets. Selivanov [Sel06, and references there] and also Becher and Grigorieff [BG15] studied continuous reducibility in non Hausdorff spaces, where the situation is in general much less satisfactory than in the case of the Baire space. In search for a useful notion of hierarchy outside Polish zero-dimensional spaces, Motto Ros, Schlicht, and Selivanov [MSS15] consider reductions by discontinuous functions. For example they obtain that the Borel subsets of the real line are well-founded with antichains of size at most 2 when quasi-ordered 0 by reducibility via functions 푓 ∶ ℝ → ℝ such that for every 퐴 ∈ 횺3(ℝ) we have −1 0 0 푓 (퐴) ∈ 횺3. They leave open the question whether 횺3 can be replaced by 0 횺2 in the above statement. Arguably one defect of this qo is that it does not refine the low level Borel classes, nor does it respect the Hausdorff hierarchy 0 of the 횫2. Instead of considering reduction by discontinuous functions, we propose to keep continuity but to release the second concept at stake, namely that of function. In the abstract, our first remark is that total relations account perfectly for the idea of reducibility and in fact generalise the framework of reductions as functions. The notion of continuity for relations that fits our purpose is called relative continuity. It relies on the simple and fundamental concept of admissible representation of a topological space which is the starting point of the development of computable analysis from the point of view of Type-2 theory of effectivity [Wei00]. The basic idea is to represent the points of a topological space 풳 by means of infinite sequences of natural numbers. Given such a representation of 풳, i.e. a partial surjective function 휌 ∶⊆ 휔휔 → 풳, an 훼 ∈ 휔휔 is a name for a point 푥 ∈ 풳 when 휌(훼) = 푥. A function 푓 ∶ 풳 → 풳 is then said to be relatively continuous (resp. computable) with respect to 휌 if the function 푓 is continuous (or computable) in the 휌-names, i.e. there exists a continuous (resp. computable) 퐹 ∶ dom 휌 → dom 휌 such that 푓 ∘ 휌 = 휌 ∘ 퐹 . Of course the notion of relatively continuous function depends on the considered representation. How- ever, for every second countable 푇0 space 풳 there exists – up to equivalence – a greatest representation (cf. Theorem 5.17) among the continuous ones, called an admissible representation of 풳. The importance of admissible representations resides in the following fact (cf. Theorem 5.24): for an admissible representation 휌 of 풳, a function 푓 ∶ 풳 → 풳 is relatively continuous with respect to 휌 if and only if 푓 is continuous. Notice however that in general as

8 1 Introduction long as the representation is not injective, many continuous transformations of the names exist which do not induce a map on the space 풳. Indeed different names 훼, 훽 of some point 푥 can be sent by a continuous function 퐹 onto names 퐹 (훼), 퐹 (훽) representing different points, i.e. 휌(퐹 (훼)) ≠ 휌(퐹 (훽)). Such transformations are called relatively continuous relations (cf. Definition 5.29) and they were first investigated in a systematic manner by Brattka and Hertling [BH94]. We propose to consider reducibility by total relatively continuous relations. When we fix an admissible representation 휌 of a second countable 푇0 space 풳, it is natural to think of reductions by relatively continuous relations as ‘reductions in the names’: if 퐴, 퐵 ⊆ 풳, then 퐴 reduces to 퐵, in symbols 퐴 ≼W 퐵, if and only if there exists a continuous function 퐹 from the names to the names such that for every name 훼, 휌(훼) ∈ 퐴 ↔ 휌(퐹 (훼)) ∈ 퐵. In other words, for every point 푥 and every name 훼 for 푥, 퐹 (훼) is the name of a point that belongs to 퐵 if and only if 푥 belongs to 퐴. We wish to mention that in 1981 Tang [Tan81] worked with an admissible representation of the Scott domain 풫휔 and studied on this particular space the exact same notion of reduction that we propose here in a more general setting. But firstly, this study is antecedent to the introduction by Kreitz and Weihrauch [KW85] of the concept of admissible representation and Tang does not notice that his representation of 풫휔 is admissible. This remark is indeed important since it allows one to see that his notion of reduction is actually topological, namely it depends only on the topology of the space 풫휔. Secondly, even though his paper is often cited, no author seem to notice his particular approach to reducibility on 풫휔. To confront the quasi-order ≼W of reducibility by relatively continuous relations to our expectations, we show the following results. Firstly, we show that reducibility by relatively continuous relations is a generalisation of Wadge reducibility outside zero-dimensional spaces. Proposition 1.6. On every zero-dimensional space, the reducibility by relatively continuous relations coincides with the continuous reducibility. Notice however that using a result of Schlicht [Sch] we show that it differs from the continuous reducibility in every separable metrisable space that is not zero-dimensional. Secondly, using a result by Saint Raymond [Sai07] extended by de Brecht [deB13] we obtain that that reducibility by relatively continuous relations refines the classical hierarchies of Borel and Hausdorff–Kuratowski.

Proposition 1.7. Let 풳 be a second countable 푇0 spaces and 퐴 and 퐵 be subsets of 풳. For every 1 ⩽ 훼, 휉 < 휔1,

9 1 Introduction

0 0 (i) if 퐵 ∈ 횺훼(풳) and 퐴 ≼W 퐵, then 퐴 ∈ 횺훼(풳),

0 0 (ii) if 퐵 ∈ 퐷휉(횺훼(풳)) and 퐴 ≼W 퐵, then 퐴 ∈ 퐷휉(횺훼(풳)).

Finally, we show that the quasi-order ≼W is as well behaved on the Borel sets of a very large class of second countable 푇0 spaces as the Wadge quasi-order is on the Borel subsets of the Baire space. The use of Borel determinacy naturally leads us to define the class of Borel representable spaces, which contains every Borel subspace of the Scott domain 풫휔, and in particular every Borel subspace of a Polish space.

Theorem 1.8. Let 풳 be a Borel representable space. Then the reducibility by relatively continuous relations ≼W is well-founded on the Borel subsets of 풳. Moreover the Wadge Lemma holds, namely for every Borel subset 퐴 and 퐵 of 풳 ∁ either 퐴 ≼W 퐵 or 퐵 ≼W 퐴 . As in the case of the Baire space, this structural result depends on the determinacy of the games under consideration. In particular, under the Axiom of Determinacy, the above theorem extends to all subsets of every second countable 푇0 space.

1.3 Organisation of the thesis

Chapter 2: Sequences in sets and orders Several articles – notably [Mil85; Kru72; Sim85; Lav71; Lav76; For03] – contains valuable introductory material to the theory of better-quasi-orders. However, a book entitled ‘Introduction to better-quasi-order theory’ is yet to be written. This chapter represents our attempt to give the motivated introduction to the deep definition of Nash- Williams we wished we had when we began studying the theory two years before. In Section 2.1 we prove a large number of characterisations of well-quasi- orders, all of them are folklore except the one stated in Proposition 2.14 which benefits from both an order-theoretical and a topological flavour. We make our way towards the definition of better-quasi-orders in Section 2.2. One of the difficulties we encountered when we began studying better-quasi- order is due to the existence of two main different definitions – obviously equivalent to experts – and along with them two different communities, the graph theorists and the descriptive set theorists, who only rarely cite each other in their contributions to the theory. The link between the original approach of

10 1 Introduction

Nash-Williams (graph theoretic) with that of Simpson (descriptive set theoretic) is merely mentioned by Argyros and Todorčević [AT05] alone. We present basic observations in order to remedy this situation in Subsection 2.2.3. Build- ing on an idea due to Forster [For03], we introduce the definition of better- quasi-order in a new way, using insight from one of the great contributions of descriptive set theory to better-quasi-order theory, namely the use of games and determinacy. Finally in Section 2.3 we put the definition of better-quasi-order into perspective. This last section contains original material which have not yet been published by the author.

Chapter 3: Sequences in spaces Building on the previous chapter, we study super-sequences in metric spaces. After making some simple observations on Cauchy sequences, we define Cauchy super-sequences in Section 3.1 and collect some basic facts about the closure of a front inside 2휔. The main result of this chapter is that any super-sequence into a compact metric space 풳 admits a Cauchy sub-super-sequence. This general result actually follows easily from the particular case where 풳 is the Cantor space. Our reason to focus on the Cantor space lies in the fact that uniform continuity admits of a nice characterisation in the zero-dimensional setting as showed in Section 3.2. In particular the uniform structure of a front 퐹 essentially consists in a distinguished countable Boolean algebra of subsets of 퐹 (a characterisation of which is given in Proposition 3.20), that we call the blocks of the front. In Section 3.3 we prove that any countable family of subsets of a front can be turned into blocks by eventually going to a sub-front in Theorem 3.24. From this combinatorial result we deduce that every super-sequence in 2휔 admits a Cauchy super-sequence. Of course, when 풳 is a complete metric space and 푓 ∶ 퐹 → 풳 is a Cauchy super-sequence, then 푓 extends to a continuous map 푓̄ ∶ 퐹 → 풳, where 퐹 is the topological closure of 퐹 inside 2휔. We study continuous map from the closure 퐹 of a front into an arbitrary topological space in Section 3.4. In particular we define a certain ‘normal form’ for the continuous functions 푓 ∶ 퐹 → 풴 from the closure of a front 퐹 into a topological space 풴 and we prove that this normal form can always be achieved by a restriction to some 퐻 where 퐻 is a sub-front of 퐹 . Importantly, these results are applied in the next chapter to prove the conjecture suggested by Pouzet [Pou78]. While the exposition given in this chapter is new, the results are published in an article [CP14] by the author and R. Carroy in Fundamenta Mathematicae.

11 1 Introduction

Chapter 4: The ideal space of a well-quasi-order The main result of this chapter is the proof of a conjecture made by Pouzet [Pou78] which relates the bqo character of a given wqo with the bqo character of the remainder of the ideal completion of the wqo. Our approach relies on the fact that the ideal completion of a wqo is actually a compactification. This can be explained by the coincidence in the case of a wqo of the ideal completion with two other important completions of a quasi-order, the properties of which combine to yield what we call the ideal space of a wqo. We do not attempt to be comprehensive on the tentacular topic of completions of partial orders. Most of the results of Section 4.1 certainly belongs to the folklore but while most authors focus mainly on lattice theoretic or domain theoretic aspects, we concentrate on the wqo property. In Subsection 4.1.2 we supply the definition of the so-called Cauchy ideal completion of a partial order which is studied by Erné and Palko [EP98] with a different approach. We give a characterisation of the partial orders inwhich the Cauchy ideals coincide with the ideals. They are the partial orders which enjoy the so-called property 푀 – well-known in domain theory. Notably the two notions coincide in the case of a wqo, since every wqo trivially satisfies property 푀. Moreover for the partial orders with property 푀, we show that the ideal completion when equipped with the Lawson topology coincides with the Cauchy ideal completion. Next we present the Cauchy ideal completion of a partial order 푃 as the Priestley dual of a certain lattice of subsets of 푃 . Following Bekkali, Pouzet, and Zhani [BPZ07] we view this as a particular case of a duality result relating the ‘taking of the topological closure’ with the ‘algebraic generation of a lattice’. We also provide a new proof of this duality result. In particular, it turns out that the Cauchy ideal completion of a wqo is the Priestley dual of the lattice of downsets. This observation leads us in Subsection 4.1.4 to consider the profinite completion of a partial order. This is also the Nachbin order-compactification of the partial order considered with the discrete topology. We see the coincidence of these various completions in the case of a wqo as a sign of the importance of the space of ideals of a wqo. We then utilise the results of Chapter 3 to prove Pouzet’s conjecture in Section 4.2. A slightly different proof was published by the author andR. Carroy [CP14]. We close this chapter by discussing some applications of Pouzet’s conjecture in Section 4.3. We notably obtain as a corollary that an interval order is wqo if and only if it is bqo, as observed by Pouzet and Sauer [PS06].

12 1 Introduction

Chapter 5: A Wadge Hierarchy for second countable spaces This chapter is based on an article [Peq15] published by the author in Archive for Mathem- atical Logic. The fact that the Wadge quasi-order is well-founded on Borel subsets of 휔휔 relies on the determinacy of certain infinite games and this result is actually best seen as an immediate corollary of a theorem on bqos obtained by van Engelen, Miller, and Steel [vEMS87]. Elaborating on Chapter 2, we present in Section 5.1 a slight generalisation of this theorem (cf Theorems 5.7 and 5.9) in a way which makes it appear as an extension of the idea underlying the very definition of bqo. From these results, we get that the quasi-order of continuous reducibility on the Borel subsets of any zero-dimensional Luzin3 space 풳 is a wqo – in fact a bqo – which satisfies the Wadge Lemma, namely for every Borel 퐴, 퐵 ⊆ 풳 either 퐴 ⩽W 퐵 or 퐵 ⩽W 풳 ∖ 퐴. In particular antichains have size at most 2. The main idea of this chapter is to generalise the Wadge quasi-order to a large class of spaces while maintaining the nice properties it enjoys on the Borel subsets of the Baire space. To do this we move from reductions by continuous functions to reductions by ‘continuous’ relations. To begin with, we observe in Section 5.2 that total relations account perfectly for the idea of reducibility in the abstract and in fact generalise the framework of reductions as functions. The notion of continuity for relations that fits our purpose is called relative continuity. It relies on the concept of admissible representation of a topological space. While this concept is fundamental to Type-2 Theory of Effectivity (see the textbook by Weihrauch [Wei00]), we do not expect our reader to be familiar with the simple and interesting underpinning of this approach to computable analysis. We therefore review the basic definitions and provide proofs for his convenience in Section 5.3. This Section ends with the definition of the quasi- order ≼W of reducibility by relatively continuous relations. We prove in Section 5.4 that the quasi-order ≼W refines the classical hierarchies of Borel and Hausdorff–Kuratowski. We define a general reduction game for represented spaces in Section 5.5 as a simple adaptation of the game we used in Section 5.1. This allows us to show that the quasi-order ≼W satisfies the Wadge Lemma on Borel subsets of Borel representable spaces. Also, moving from continuous functions to relatively continuous relations we extend our version of the theorem by van Engelen, Miller, and Steel [vEMS87] from Luzin zero-dimensional spaces to all Borel representable spaces. This yields in particular that the reducibility by relatively continuous relations is well-founded – in fact bqo – on the Borel

3Luzin spaces are also called Borel absolute spaces.

13 1 Introduction subsets of every Borel representable space. Finally in Section 5.6 we exemplify the difference between the continuous reducibility and the reducibility by relatively continuous relations in two major examples: the real line ℝ and the Scott domain 풫휔.

14 2 Sequences in sets and orders

Sequences play an important rôle in both the theory of topological spaces and the theory of orders. A sequence in a set 퐸 is simply a map from the set 휔 of natural numbers into 퐸. A subsequence of a sequence 푓 ∶ 휔 → 퐸 is a restriction

푓↾푋 ∶ 푋 → 퐸 of 푓 to some infinite subset 푋 of 휔. In this chapter we focus on sequences and their generalisation, super-sequences, in the context of sets and ordered sets. This leads us from well-quasi-orders to better-quasi-orders.

2.1 Well-quasi-orders

A reflexive and transitive binary relation ⩽ on a set 푄 is called a quasi-order (qo, also preorder). As it is customary, we henceforth make an abuse of terminology and refer to the pair (푄, ⩽) simply as 푄 when there is no danger of confusion. Moreover when it is necessary to prevent ambiguity we use a subscript and write ⩽푄 for the binary relation of the quasi-order 푄. The notion of quasi-order is certainly the most general mathematical concept of ordering. Two elements 푝 and 푞 of a quasi-order 푄 are equivalent, in symbols 푝 ≡ 푞, if both 푝 ⩽ 푞 and 푞 ⩽ 푝 hold. It can very well happen that 푝 is equivalent to 푞 while 푝 is not equal to 푞. This kind of situation naturally arises when one considers for example the quasi-order of embeddability among a certain class of structures. Examples of pairs of structures which mutually embed into each other while being distinct, or even non isomorphic, abound in mathematics. Every quasi-order possess an associated strict relation, denoted by <, defined by 푝 < 푞 if and only if 푝 ⩽ 푞 and 푞 ⩽̸ 푝 – equivalently 푝 ⩽ 푞 and 푝 ≢ 푞. We say two elements 푝 and 푞 are incomparable, when both 푝 ⩽̸ 푞 and 푞 ⩽̸ 푝 hold, in symbols 푝 | 푞. A map 푓 ∶ 푃 → 푄 between quasi-orders is order-preserving (also isotone) ′ ′ if whenever 푝 ⩽푃 푝 holds in 푃 we have 푓(푝) ⩽푄 푓(푝 ) in 푄. An embedding ′ ′ is a map 푓 ∶ 푃 → 푄 such that for every 푝 and 푝 in 푃 , 푝 ⩽푃 푝 if and ′ only if 푓(푝) ⩽푄 푓(푝 ). Notice that an embedding is not necessarily injective. An embedding 푓 ∶ 푃 → 푄 is called an equivalence1 provided it is essentially

1Viewing quasi-orders as categories in the obvious way, this notion of equivalence coincides with the one used in category theory.

15 2 Sequences in sets and orders

surjective, i.e. for every 푞 ∈ 푄 there exists 푝 ∈ 푃 with 푞 ≡푄 푓(푝). We say that two quasi-orders 푃 and 푄 are equivalent if there exists an equivalence from 푃 to 푄 – by the axiom of choice this is easily seen to be an equivalence relation on the class of quasi-orders. Notice that every set 푋 quasi-ordered by the full relation 푋 × 푋 is equivalent to the one point quasi-order. In contrast, by an isomorphism 푓 from 푃 to 푄 we mean a bijective embedding 푓 ∶ 푃 → 푄. Of course, a set 푋 quasi-ordered with the full relation 푋 × 푋 is never isomorphic to 1 except when 푋 contains exactly one element. In the sequel we study quasi-orders up to equivalence, namely only properties of quasi-orders which are preserved by equivalence are considered. A quasi-order 푄 is called a partial order (po, also poset) provided the relation ⩽ is antisymmetric, i.e. 푝 ≡ 푝 implies 푝 = 푞 – equivalent elements are equal. Notice that an embedding between partial orders is necessarily injective. Moreover if 푃 and 푄 are partial orders and 푓 ∶ 푃 → 푄 is an equivalence, then 푓 is an isomorphism. We also note that in a partial order the associated strict order can also be defined by 푝 < 푞 if and only if 푝 ⩽ 푞 and 푝 ≠ 푞. Importantly, every quasi-order 푄 admits up to isomorphism a unique equivalent partial order, its equivalent partial order, which can be obtained as the quotient of 푄 by the equivalence relation 푝 ≡ 푞. Even though most naturally occurring examples and constructions are only quasi-orders, one can always think of the equivalent partial order. The study of quasi-orders therefore really amounts to the study of partial orders.

2.1.1 Good versus bad sequences We let 휔 = {0, 1, 2, …} be the set of natural numbers. We use the set theoretic definitions 0 = ∅ and 푛 = {0, … , 푛 − 1}, so that the usual order on 휔 coincides with the membership relation. The equality and the usual order on 휔 give rise to the following distinguished types of sequences into a quasi-order.

Definitions 2.1. Let 푄 be a quasi-order.

(1) An infinite antichain is a map 푓 ∶ 휔 → 푄 such that for all 푚, 푛 ∈ 휔, 푚 ≠ 푛 implies 푓(푚) | 푓(푛).

(2) An infinite descending chain, or an infinite decreasing sequence in 푄, is a map 푓 ∶ 휔 → 푄 such that for all 푚, 푛 ∈ 휔, 푚 < 푛 implies 푓(푚) > 푓(푛).

(3) A perfect sequence, is a map 푓 ∶ 휔 → 푄 such that for all 푚, 푛 ∈ 휔 the relation 푚 ⩽ 푛 implies 푓(푚) ⩽ 푓(푛). In other words, 푓 is perfect if it is order-preserving from (휔, ⩽) to (푄, ⩽).

16 2 Sequences in sets and orders

(4) A bad sequence is a map 푓 ∶ 휔 → 푄 such that for all 푚, 푛 ∈ 휔, 푚 < 푛 implies 푓(푚) ⩽̸ 푓(푛).

(5) A good sequence is a map 푓 ∶ 휔 → 푄 such that there exist 푚, 푛 ∈ 휔 with 푚 < 푛 and 푓(푚) ⩽ 푓(푛). Hence a sequence is good exactly when it is not bad.

For any given set 푋, we denote by [푋]2 the set of pairs {푥, 푦} for distinct 푥, 푦 ∈ 푋. By Ramsey’s Theorem2 [Ram30] whenever [휔]2 is partitioned into 2 푃0 and 푃1 there exists an infinite subset 푋 of 휔 such that either [푋] ⊆ 푃0, 2 or [푋] ⊆ 푃1. Proposition 2.2. For a quasi-order 푄, the following conditions are equivalent.

(W1) 푄 has no infinite descending chain and no infinite antichain;

(W2) there is no bad sequence in 푄;

(W3) every sequence in 푄 admits a perfect subsequence.

Proof. (W1)→(W2): By contraposition, suppose that 푓 ∶ 휔 → 푄 is a bad 2 sequence. Partition [휔] into 푃0 and 푃1 with

2 ⩾̸ 푃0 = {{푚, 푛} ∈ [휔] ∣ 푚 < 푛 and 푓(푚) 푓(푛)}.

By Ramsey’s Theorem, there exists an infinite subset 푋 of integers with 2 2 either [푋] ⊆ 푃0, or [푋] ⊆ 푃1. In the first case 푓 ∶ 푋 → 푄 is an infinite antichain and in the second case 푓 ∶ 푋 → 푄 is an infinite descending chain.

(W2)→(W1): Notice that an infinite antichain and an infinite descending chain are two examples of a bad sequence.

2 (W2)↔(W3): Let 푓 ∶ 휔 → 푄 be any sequence in 푄. We partition [휔] in 푃0 and 푃1 with

2 ⩽̸ 푃0 = {{푚, 푛} ∈ [휔] ∣ 푚 < 푛 and 푓(푚) 푓(푛)}.

By Ramsey’s Theorem, there exists an infinite subset 푋 of integers such 2 2 that [푋] ⊆ 푃0 or [푋] ⊆ 푃1. The first case yields a bad subsequence. The second case gives a perfect subsequence.

2Nash-Williams’ generalisation of Ramsey’s Theorem is stated and proved as Theorem 2.36.

17 2 Sequences in sets and orders

Definition 2.3. A quasi-order 푄 is called a well-quasi-order (wqo) when one of the equivalent conditions of the previous proposition is fulfilled. A quasi- order with no infinite descending chain is said tobe well-founded.

The notion of wqo is a frequently discovered concept, for an historical account of its early development we refer the reader to the excellent article by Kruskal [Kru72]. Using Proposition 2.2 and the Ramsey’s Theorem for pairs, one easily proves the following basic closure properties of the class of wqos.

Proposition 2.4.

′ (i) If (푄, ⩽푄) is wqo and 푃 ⊆ 푄, then (푃 , ⩽푃 ) is wqo, where 푝 ⩽푃 푝 if ′ ′ and only if 푝, 푝 ∈ 푃 and 푝 ⩽푄 푝 .

(ii) If (푃 , ⩽푃 ) and (푄, ⩽푄) are wqo, then 푃 × 푄 quasi-ordered by

′ ′ ′ ′ (푝, 푞) ⩽푃 ×푄 (푝 , 푞 ) ⟷ 푝 ⩽푃 푝 and 푞 ⩽푄 푞

is wqo.

(iii) If (푃 , ⩽ ) is a partial order and (푄 , ⩽ ) is a quasi-order for every 푃 푝 푄푝 푝 ∈ 푃 , the sum ∑ 푄 of the 푄 along 푃 has underlying set the 푝∈푃 푝 푝 disjoint union {(푝, 푞) ∣ 푝 ∈ 푃 and 푞 ∈ 푄푝} and is quasi-ordered by

(푝, 푞) ⩽ (푝′, 푞′) ⟷ either 푝 = 푝′ and 푞 ⩽ 푞′, or 푝 < 푝′. 푄푝

If 푃 is wqo and each 푄 is wqo, then ∑ 푄 is wqo. 푝 푝∈푃 푝 (iv) If 푄 is wqo and there exists a map 푔 ∶ 푃 → 푄 such that for all 푝, 푝′ ∈ 푃 푔(푝) ⩽ 푔(푝′) → 푝 ⩽ 푝′, then 푃 is wqo.

(v) If 푃 is wqo and there is a surjective and monotone map ℎ ∶ 푃 → 푄, then 푄 is wqo.

2.1.2 Subsets and downsets Importantly, a wqo can be characterised in terms of its subsets.

Definitions 2.5. Let 푄 be a quasi-order.

18 2 Sequences in sets and orders

(1) A subset 퐷 of 푄 is a downset, or an initial segment, if 푞 ∈ 퐷 and 푝 ⩽ 푞 implies 푝 ∈ 퐷. For any 푆 ⊆ 푄, we write ↓ 푆 for the downset generated by 푆 in 푄, i.e. the set {푞 ∈ 푄 ∣ ∃푝 ∈ 푆 푞 ⩽ 푝}. We also write ↓ 푝 for ↓{푝}. We denote by 풟(푄) the po of downsets of 푄 under inclusion. (2) We give the dual meaning to upset, ↑ 푆 and ↑ 푞 respectively. (3) An upset 푈 is said to to be finitely generated, or to admit a finite basis, if there exists a finite 퐹 ⊆ 푈 such that 푈 = ↑ 퐹 . We say that 푄 has the finite basis property if every upset of 푄 admits a finite basis. (4) A downset 퐷 ∈ 풟(푄) is said to be finitely bounded, if there exists a finite set 퐹 ⊆ 푄 with 퐷 = 푄 ∖ ↑ 퐹 . We let 풟fb(푄) be the set of finitely bounded downsets partially ordered by inclusion. (5) We turn the power-set of 푄, denoted 풫(푄), into a qo by letting 푋 ⩽ 푌 if and only if ∀푝 ∈ 푋 ∃푞 ∈ 푌 푝 ⩽ 푞, this is sometimes called the domination quasi-order. We let 풫 (푄) be the the set of countable <ℵ1 subsets of 푄 with the quasi-order induced from 풫(푄). Since 푋 ⩽ 푌 if and only if ↓ 푋 ⊆ ↓ 푌 , the equivalent partial order of 풫(푄) is 풟(푄) and the quotient map is given by 푋 ↦ ↓ 푋. The notion of well-quasi-order should be thought of as a generalisation of the notion of well-ordering beyond linear orders. Recall that a po 푃 is a linear order if for every 푝 and 푞 in 푃 , either 푝 ⩽ 푞 or 푞 ⩽ 푝.A well-ordering is (traditionally, the associated strict relation < of) a partial order that is both linearly ordered and well-founded. Observe that a linearly ordered 푃 is well-founded if and only if the initial segments of 푃 are well-founded under inclusion. Considering for example the po (휔, =), one directly sees that a partial order 푃 can be well-founded while the initial segments of 푃 (here 풫(휔)) are not well-founded under inclusion. However a qo 푄 is wqo if and only if the initial segments of 푄 are well-founded under inclusion. Proposition 2.6. A quasi-order 푄 is a wqo if and only if one of the following equivalent conditions is fulfilled: (W4) 푄 has the finite basis property, (W5) (풫(푄), ⩽) is well-founded, (W6) (풫 (푄), ⩽) is well-founded, <ℵ1

19 2 Sequences in sets and orders

(W7) (풟(푄), ⊆) is well-founded,

(W8) (풟fb(푄), ⊆) is well-founded. Proof. (W2)→(W4): By contraposition, suppose 푆 ∈ 풰(푄) admits no finite basis. Since ∅ = ↑ ∅, 푆 ≠ ∅. By dependent choice, we can show the existence of a bad sequence 푓 ∶ 휔 → 푄. Choose 푓(0) ∈ 푆 and suppose that 푓 is defined up to some 푛 > 0. Since ↑{푓(0), … 푓(푛)} ⊂ 푆 we can choose some 푓(푛 + 1) inside 푆 ∖ ↑{푓(0), … 푓(푛)}.

(W4)→(W5): By contraposition, suppose that (푋푛)푛∈휔 is an infinite descending chain in 풫(푄). Then for each 푛 ∈ 휔 we choose 푞푛 ∈ ↓ 푋푛 ∖ ↓ 푋푛+1. Then {푞푛 ∣ 푛 ∈ 휔} has no finite basis. Indeed for all 푛 ∈ 휔 we have 푞푛+1 ∉ ↑{푞푖 ∣ 푖 ⩽ 푛}, otherwise 푞푖 ⩽ 푞푛+1 ∈ ↓ 푋푛+1 ⊆ ↓ 푋푖+1 for some 푖 ⩽ 푛, a contradiction.

(W5)→(W6): Obvious.

(W6)→(W2): By contraposition, if (푞푛)푛∈휔 is a bad sequence in 푄, then 푃푛 = {푞 ∣ 푛 ⩽ 푘} is an infinite descending chain in 풫 (푄) since whenever 푘 <ℵ1 ⩽̸ 푚 < 푛 we have 푞푚 ∈ 푃푚 and 푞푚 푞푘 for every 푘 ⩾ 푛. (W5)→(W7): By contraposition, notice that an infinite descending chain in (풟(푄), ⊆) is also an infinite descending chain in (풫(푄), ⩽).

(W7)→(W8): Obvious.

(W8)→(W2): By contraposition, if 푓 ∶ 휔 → 푄 is a bad sequence, then 푛 ↦ 퐷푛 = 푄 ∖ ↑{푓(푖) ∣ 푖 ⩽ 푛} is an infinite descending chain in 풟fb(푄).

2.1.3 Regular sequences A monotone decreasing sequence of ordinals is, by well-foundedness, eventually constant. The limit of such a sequence exists naturally, and is simply its minimum. In general the limit of a sequence (훼푖)푖∈휔 of ordinals may not exist, however any sequence of ordinals admits a limit superior. Indeed, define the sequence 훽 = sup 훼 = ⋃ 훼 , then (훽 ) is decreasing and hence admits a limit. 푖 푗⩾푖 푗 푗⩾푖 푗 푖 푖∈휔 We say that (훼 ) is regular if the limit superior and the supremum ⋃ 훼 푖 푖∈휔 푖∈휔 푖 of (훼푖)푖∈휔 coincide. This is equivalent to saying that for every 푖 ∈ 휔 there exists 푗 > 푖 with 훼푖 ⩽ 훼푗. By induction one shows that this is in turn equivalent to saying that for all 푖 ∈ 휔 the set {푗 ∈ 휔 ∣ 푖 < 푗 and 훼푖 ⩽ 훼푗} is infinite.

20 2 Sequences in sets and orders

Notation 2.7. For 푛 ∈ 휔 and 푋 an infinite subset of 휔 let us denote by 푋/푛 the final segment of 푋 given by {푘 ∈ 푋 ∣ 푘 > 푛}.

We generalise the definition of regular sequences of ordinals to sequences in quasi-orders as follows.

Definition 2.8. Let 푄 be a qo. A regular sequence is a map 푓 ∶ 휔 → 푄 such that for all 푛 ∈ 휔 the set {푘 ∈ 휔/푛 ∣ 푓(푛) ⩽ 푓(푘)} is infinite.

Here is a characterisation of wqo in terms of regular sequences which exhibits another property of well-orders shared by wqos.

Proposition 2.9. Let 푄 be a qo. Then 푄 is wqo if and only if one of the following equivalent conditions holds:

(W9) Every sequence in 푄 admits a regular subsequence.

(W10) For every sequence 푓 ∶ 휔 → 푄 there exists 푛 ∈ 휔 such that the restriction 푓 ∶ 휔/푛 → 푄 is regular.

Proof. (W7)→(W10): For 푓 ∶ 휔 → 푄 we let 푓̂ ∶ 휔 → 풟(푄) be defined by 푓(푛)̂ = ↓{푓(푘) ∣ 푛 ⩽ 푘 < 휔}. Then clearly if 푚 < 푛 then 푓(푚)̂ ⊇ 푓(푛)̂ . The partial order 풟(푄) being well-founded by (W7), there exists 푛 ∈ 휔 such that for every 푚 > 푛 we have 푓(푛)̂ = 푓(푚)̂ . This 푛 is as desired. Indeed, if 푘 > 푛 then for every 푙 > 푘 we have 푓(푘) ∈ 푓(푘)̂ = 푓(푙)̂ and so there exists 푗 ⩾ 푙 with 푓(푘) ⩽ 푓(푙).

(W10)→(W9): Obvious.

(W9)→(W2): By contraposition, if 푓 ∶ 휔 → 푄 is a bad sequence, then every subsequence of 푓 is bad. Clearly a bad sequence 푓 ∶ 휔 → 푄 is not regular since for every 푛 ∈ 휔 the set {푘 ∈ 휔/푛 ∣ 푓(푛) ⩽ 푓(푘)} is empty. Hence a bad sequence admits no regular subsequence.

2.1.4 Sequences of subsets In this subsection we give a new characterisation of wqos which enjoys both a topological and an order-theoretical flavour. So far, we have considered 풟(푄) as partially ordered set for inclusion. But 풟(푄) also admits a natural topology which turns it into a compact Hausdorff 0-dimensional space. Consider 푄 as a discrete topological space, and form the product space 2푄, whose underlying set is identified with 풫(푄). This product

21 2 Sequences in sets and orders space, sometimes called generalised Cantor space, admits as a basis the clopen sets of the form

푁(퐹 , 퐺) = {푋 ⊆ 푄 ∣ 퐹 ⊆ 푋 and 푋 ∩ 퐺 = ∅}, for finite subsets 퐹 , 퐺 of 푄. For 푞 ∈ 푄, we write ⟨푞⟩ instead of 푁({푞}, ∅) for the clopen set {푋 ⊆ 푄 ∣ 푞 ∈ 푋}. Note that ⟨푞⟩∁ = 푁(∅, {푞}). Notice that 풟(푄) is an intersection of clopen sets,

풟(푄) = ⋂⟨푞⟩∁ ∪ ⟨푝⟩, 푝⩽푞 hence 풟(푄) is closed in 2푄 and therefore compact. Now recall that for every sequence (퐸푛)푛∈휔 of subsets of 푄 we have the usual relations ⋂ 퐸푛 ⊆ ⋃ ⋂ 퐸푗 ⊆ ⋂ ⋃ 퐸푗 ⊆ ⋃ 퐸푛. (2.1) 푛∈휔 푖∈휔 푗⩾푖 푖∈휔 푗⩾푖 푛∈휔 Moreover the convergence of sequences in 2푄 can be expressed by means of a ‘lim inf = lim sup’ property.

푄 Fact 2.10. A sequence (퐸푛)푛∈휔 converges to 퐸 in 2 if and only if

⋃ ⋂ 퐸푗 = ⋂ ⋃ 퐸푗 = 퐸. 푖∈휔 푗⩾푖 푖∈휔 푗⩾푖

Proof. Suppose that 퐸 = ⋃ ⋂ 퐸 = ⋂ ⋃ 퐸 . We show that 퐸 → 푖∈휔 푗⩾푖 푗 푖∈휔 푗⩾푖 푗 푛 퐸. Let 퐹 , 퐺 be finite subsets of 푄 with 퐸 ∈ 푁(퐹 , 퐺). Since 퐸 = ⋃ ⋂ 퐸 푖∈휔 푗⩾푖 푗 and 퐹 finite, 퐹 ⊆ 퐸 for all sufficiently large 푗. Since 퐸 = ⋂ ⋃ 퐸 and 푗 푖∈휔 푗⩾푖 푗 퐺 is finite, 퐺 ∩ 퐸푗 = ∅ for all sufficiently large 푗. It follows that 퐸푗 ∈ 푁(퐹 , 퐺) for all sufficiently large 푗, whence (퐸푛)푛 converges 퐸. 푄 Conversely, assume that 퐸푛 converges to some 퐸 in 2 . If 푞 belongs to 퐸 – i.e. 퐸 ∈ ⟨푞⟩ – then 푞 ∈ 퐸 for all sufficiently large 푗 and thus 푞 ∈ ⋃ ⋂ 퐸 . 푗 푖∈휔 푗⩾푖 푗 And if 푞 ∉ 퐸, i.e. 퐸 ∉ ⟨푞⟩, then 푞 ∉ 퐸푗 for all sufficiently large 푗 and thus 푞 ∉ ⋂ ⋃ 퐸 . Therefore by (2.1) it follows that 퐸 = ⋃ ⋂ 퐸 = 푖∈휔 푗⩾푖 푗 푖∈휔 푗⩾푖 푗 ⋂ ⋃ 퐸 . 푖∈휔 푗⩾푖 푗

Observe that if (푞푛)푛∈휔 is a perfect sequence in a qo 푄, then for every 푞 ∈ 푄 if 푞 ⩽ 푞푚 for some 푚, then 푞 ⩽ 푞푛 holds for all 푛 ⩾ 푚. Therefore by (2.1) we have ⋃ ⋂ ↓ 푞푛 = ⋃ ↓ 푞푛, 푚∈휔 푛⩾푚 푛∈휔

22 2 Sequences in sets and orders

푄 whence (↓ 푞푛)푛∈휔 converges to ↓{푞푛 ∣ 푛 ∈ 휔} in 2 by Fact 2.10. On the contrary no bad sequence (푞푛)푛∈휔 converges towards ↓{푞푛 ∣ 푛 ∈ 휔}, since for example 푞 does not belong to ⋃ ⋂ ↓ 푞 . We have obtained the following: 0 푖∈휔 푗⩾푖 푗 Fact 2.11. Let 푄 be a qo.

∞ (i) 푄 is wqo if and only if for every sequence (푞푛)푛∈휔 there exists 푁 ∈ [휔] such that (↓ 푞푛)푛∈푁 converges to ↓{푞푛 ∣ 푛 ∈ 푁} in 풟(푄).

(ii) If 푄 is wqo and (↓ 푞푛)푛∈휔 converges to some 퐷 in 풟(푄), then there is ∞ some 푁 ∈ [휔] such that 퐷 = ↓{푞푛 ∣ 푛 ∈ 푁}. Actually more is true, thanks to the following ingenious observation made by Richard Rado in the body of a proof in [Rad54].

Lemma 2.12 (Rado’s trick). Let 푄 be a wqo and let (퐷푛)푛∈휔 be a sequence in 풟(푄). Then there exists an infinite subset 푁 of 휔 such that

⋃ ⋂ 퐷푗 = ⋃ 퐷푛, 푖∈푁 푗∈푁/푖 푛∈푁 and so the subsequence (퐷 ) converges to ⋃ 퐷 in 풟(푄). 푗 푗∈푁 푛∈푁 푛 Proof. Towards a contradiction suppose that for all infinite 푁 ⊆ 휔 we have

⋃ ⋂ 퐷푗 ⊂ ⋃ 퐷푛. (2.2) 푖∈푁 푗∈푁/푖 푛∈푁

We define an infinite descending chain (퐸푖)푖∈휔 in 풟(푄). But to do so we recursively define a sequence (푁푘)푘∈휔 of infinite subsets of 휔 and a sequence (푞푘)푘∈휔 in 푄 such that

(a) 푁0 = 휔 and 푁푘 ⊇ 푁푘+1 for all 푘 ∈ 휔.

(b) 푞푘 ∈ ⋃ 퐷푗 and 푞푘 ∉ ⋃ 퐷푗. 푗∈푁푘 푗∈푁푘+1

Suppose we have defined 푁0, … , 푁푘 and 푞0, … 푞푘. By (2.2) we have

⋃ 퐷푛 ⊈ ⋃ ⋂ 퐷푗,

푛∈푁푘 푖∈푁푘 푗∈푁푘/푖 so we can let 푛0 ∈ 푁푘 and 푞푘 ∈ 퐷푛 be such that 푞푘 ∉ ⋃ ⋂ 퐷푗. 0 푖∈푁푘 푗∈푁푘/푖 Then for all 푖 in 푁 let 푗 ∈ 푁 /푖 be minimal such that 푞 ∉ 퐷 . Setting 푘 푖 푘 푘 푗푖

23 2 Sequences in sets and orders

푛 = 푗 and 푛 = 푗 , we obtain an infinite set 푁 = {푛 , 푛 , 푛 , …} 1 푛0 푖+1 푛푖 푘+1 0 1 2 which satisfies 푞 ∈ 퐷 ⊆ ⋃ 퐷 and 푞 ∉ ⋃ 퐷 . 푘 푛0 푗 푘 푗 푗∈푁푘 푗∈푁푘+1

Now we define 퐸푘 = ⋃ 퐷푗. The sequence (퐸푘)푘∈휔 is an infinite descend- 푗∈푁푘 ing chain in 풟(푄), contradicting the fact that 푄 is wqo. For the second statement, observe that if 푁 is an infinite subset of 휔 satisfying the statement of the lemma, then by (2.1) we have

⋃ ⋂ 퐷푗 = ⋂ ⋃ 퐷푗 = ⋃ 퐷푛, 푖∈푁 푗∈푁/푖 푖∈푁 푗∈푁/푖 푛∈푁 and so by Fact 2.10 we get that (퐷 ) converges to ⋃ 퐷 in 풟(푄). 푗 푗∈푁 푛∈푁 푛 Hence if 푄 is wqo, then every sequence in 풟(푄) admits a subsequence which converges to its union. Of course the converse also holds.

Lemma 2.13. If (퐷푛)푛∈휔 is an infinite descending chain in 풟(푄), then there is no infinite subset 푁 of 휔 such that ⋃ 퐷 = ⋃ ⋂ 퐷 . 푛∈푁 푛 푖∈푁 푗∈푁/푖 푗 Proof. Since any subsequence of an infinite descending chain is again an infinite descending chain, it is enough to show that if (퐷푛)푛∈휔 is an infinite descending chain in 풟(푄) then ⋃ 퐷 ⊈ ⋃ ⋂ 퐷 . Pick any 푞 ∈ 퐷 ∖ 퐷 . Then 푛∈휔 푛 푖∈휔 푗∈휔/푖 푗 0 1 since 퐷푗 ⊆ 퐷1 for all 푗 ⩾ 1 and 퐷1 is a down set, we get 푞 ∉ 퐷푗 for all 푗 ⩾ 1. It follows that 푞 ∉ ⋃ ⋂ 퐷 . 푖∈휔 푗∈휔/푖 푗 This leads to our last characterisation of wqo: Proposition 2.14. Let 푄 be a qo. Then 푄 is wqo if and only if

(W11) Every sequence (퐷푛)푛∈휔 in 풟(푄) admits a subsequence (퐷푛)푛∈푁 which converges to ⋃ 퐷 . 푛∈푁 푛

2.2 Better-quasi-orders

2.2.1 Towards better As we have seen in Proposition 2.6 a quasi-order is wqo if and only if 풫(푄) is well-founded if and only if 풟(푄) is well founded. The first example of a wqo whose powerset contains an infinite antichain was identified by Richard Rado. This wqo is the starting point of the journey towards the stronger notion of a better-quasi-order and it will stand as a crucial example in this thesis.

24 2 Sequences in sets and orders

⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋱

퐷3 • • • • • • • • • • • • • • • • • • • • • •

(0, 3)•(1, 3)•(2, 3)•

(0, 2)•(1, 2)•

(0, 1)•

Figure 2.1: Rado’s poset ℜ.

Example 2.15 ([Rad54]). Rado’s partial order ℜ is the set [휔]2, of pairs of natural numbers, partially ordered by (cf. Figure 2.1): 푚 = 푚′ and 푛 ⩽ 푛′, or {푚, 푛} ⩽ {푚′, 푛′} ⟷ { 푛 < 푚′. where by convention a pair {푚, 푛} of natural numbers is always assumed to be written in increasing order (푚 < 푛). The po ℜ is wqo. To see this, consider any map 푓 ∶ 휔 → [휔]2 and let 푓(푛) = {푓0(푛), 푓1(푛)} for all 푛 ∈ 휔. Now if 푓0 is unbounded, then there exists 푛 > 0 with 푓1(0) < 푓0(푛) and so 푓(0) ⩽ 푓(푛) in ℜ by the second clause. If 푓0 is bounded, let us assume by going to a subsequence if necessary, that 푓1 ∶ 휔 → 휔 is perfect. Then there exist 푚 and 푛 with 푚 < 푛 and 푓0(푚) = 푓0(푛) and we have 푓1(푚) ⩽ 푓1(푛), so 푓(푚) ⩽ 푓(푛) in ℜ by the first clause. In both case we find that 푓 is good, so ℜ is wqo. However the map 푛 ↦ 퐷푛 = ↓{{푛, 푙} ∣ 푛 < 푙} is a bad sequence (in fact an infinite antichain) inside 풟(ℜ). Indeed whenever 푚 < 푛 we have {푚, 푛} ∈ 퐷푚 while {푚, 푛} ∉ 퐷푛, and so 퐷푚 ⊈ 퐷푛. One natural question is now: What witnesses in a given quasi-order 푄 the fact that 풫(푄) is not wqo? It cannot always be a bad sequence, that is what the existence of Rado’s poset tells us. But then what is it?

25 2 Sequences in sets and orders

To see this suppose that (푃푛)푛∈휔 is a bad sequence in 풫(푄). Fix some 푚 ∈ 휔. Then whenever 푚 < 푛 we have 푃푚 ⊈ ↓ 푃푛 and we can choose a witness 푞 ∈ 푃푚 ∖ ↓ 푃푛. But of course in general there is no single 푞 ∈ 푃푚 that witnesses 푃푚 ⊈ ↓ 푃푛 for all 푛 > 푚. So we are forced to pick a sequence 푛 푓푚 ∶ 휔/푚 → 푄, 푛 ↦ 푞푚 of witnesses:

푛 푛 푞푚 ∈ 푃푚 and 푞푚 ∉ ↓ 푃푛, 푛 ∈ 휔/푚.

Bringing together all the sequences 푓0, 푓1,…, we obtain a sequence of sequences, naturally indexed by the set [휔]2 of pairs of natural numbers,

푓 ∶ [휔]2 ⟶ 푄 푛 {푚, 푛} ⟼ 푓푚(푛) = 푞푚. By our choices this sequence of sequences satisfies the following condition: 푛 ⩽̸ 푙 ∀푚, 푛, 푙 ∈ 휔 푚 < 푛 < 푙 → 푞푚 푞푛.

푛 푙 Indeed, suppose towards a contradiction that for 푚 < 푛 < 푙 we have 푞푚 ⩽ 푞푛. 푙 푛 푛 푛 Since 푞푛 ∈ 푃푛 we would have 푞푚 ∈ ↓ 푃푛, but we chose 푞푚 such that 푞푚 ∉ ↓ 푃푛. Let us say that a sequence of sequences 푓 ∶ [휔]2 → 푄 is bad if for every 푚, 푛, 푙 ∈ 휔, 푚 < 푛 < 푙 implies 푓({푚, 푛}) ⩽̸ 푓({푛, 푙}). We have come to the following. Proposition 2.16. Let 푄 be a qo. Then 풫(푄) is wqo if and only if there is no bad sequence of sequences into 푄. Proof. As we have seen in the preceding discussion, if 풫(푄) is not wqo then from a bad sequence in 풫(푄) we can make choices in order to define a bad sequences of sequences in 푄. Conversely, if 푓 ∶ [휔]2 → 푄 is a bad sequence of sequences, then for each 푚 ∈ 휔 we can consider the set 푃푚 = {푓({푚, 푛}) ∣ 푛 ∈ 휔/푚} consisting in th the image of the 푚 sequence. Then the sequence 푚 ↦ 푃푚 in 풫(푄) is a bad sequence. Indeed every time 푚 < 푛 we have 푓({푚, 푛}) ∈ 푃푚 while 푓({푚, 푛}) ∉ ↓ 푃푛, since otherwise there would exist 푙 > 푛 with 푓({푚, 푛}) ⩽ 푓({푛, 푙}), a contradiction with the fact that 푓 is a bad sequence of sequences. One should notice that from the previous proof we actually get that 풫(푄) is wqo if and only if 풫 (푄) is wqo. Notice also that in the case of Rado’s <ℵ1 partial order ℜ the fact that 풫(ℜ) is not wqo is witnessed by the bad sequence 푓 ∶ [휔]2 → ℜ, {푚, 푛} ↦ {푚, 푛} which is simply the identity on the underlying sets, since every time 푚 < 푛 < 푙 then {푚, 푛} ⩽̸ {푛, 푙} in ℜ. In fact, Rado’s partial order is in a sense universal as established by Richard Laver [Lav76]:

26 2 Sequences in sets and orders

Theorem 2.17. If 푄 is wqo but 풫(푄) is not wqo, then ℜ embeds into 푄.

Proof. Let 푓 ∶ [휔]2 → 푄 be a bad sequence of sequences. Partitioning the triples {푖, 푗, 푘}, 푖 < 푗 < 푘, into two sets depending on whether or not 푓({푖, 푗}) ⩽ 푓({푖, 푘}), we get by Ramsey’s Theorem an infinite set 푁 ⊆ 휔 whose triples are all contained into one of the classes. If for every {푖, 푗, 푘} ⊆ 푁 we have ⩽̸ 푓({푖, 푗}) 푓({푖, 푘}) then for any 푖 ∈ 푁 the sequence 푓({푖, 푗})푗∈푁/푖 is a bad sequence in 푄. Since 푄 is wqo, the other possibility must hold. Then partition the quadruples {푖, 푗, 푘, 푙} in 푁 into two sets according to whether or not 푓({푖, 푗}) ⩽ 푓({푘, 푙}). Again there exists an infinite subset 푀 of 푁 whose quadruples are all contained into one of the classes. If all quadruples ⩽̸ {푖, 푗, 푘, 푙} in 푀 satisfy 푓({푖, 푗}) 푓({푘, 푙}), then for any sequence ({푖푘, 푗푘})푘∈휔 of pairs in 푀 with 푗푘 < 푖푘+1 the sequence 푓({푖푘, 푗푘})푘∈휔 is bad in 푄. Since 푄 is wqo, it must be the other possibility that holds. Let 푋 = 푀 ∖ {min 푀}, then {푓({푖, 푗}) ∣ {푖, 푗} ∈ [푋]2} is isomorphic to ℜ. By the properties of 푀, we have {푖, 푗} ⩽ {푘, 푙} in ℜ implies 푓({푖, 푗}) ⩽ 푓({푘, 푙}). We show that 푓({푖, 푗}) ⩽ 푓({푘, 푙}) implies {푖, 푗} ⩽ {푘, 푙} in ℜ. Suppose {푖, 푗} ⩽̸ {푘, 푙} in ℜ, namely 푘 ⩽ 푗 and either 푖 ≠ 푘, or 푙 < 푗. If 푙 < 푗 and 푓({푖, 푗}) ⩽ 푓({푘, 푙}) then for any 푛 ∈ 푋/푗 we have 푓({푘, 푙}) ⩽ 푓({푗, 푛}) and thus 푓({푖, 푗}) ⩽ 푓({푗, 푛}) a contradiction since 푓 is bad. Suppose now that 푘 ⩽ 푗 and 푖 ≠ 푘. If 푖 < 푘 and 푓({푖, 푗}) ⩽ 푓({푘, 푙}), then 푓({푖, 푘}) ⩽ 푓({푖, 푗}) ⩽ 푓({푘, 푙}), a contradiction. Finally if 푘 < 푖 and 푓({푖, 푗}) ⩽ 푓({푘, 푙}) then for 푚 = min 푀 we have 푓({푚, 푘}) ⩽ 푓({푖, 푗}) ⩽ 푓({푘, 푙}), again a contradiction. From a heuristic viewpoint, a better-quasi-order is a well quasi-order 푄 such that 풫(푄) is wqo, 풫(풫(푄)) is wqo, 풫(풫(풫(푄))) is wqo, so on and so forth, into the transfinite. This idea will be made precise in Subsection 2.2.4, but we can already see that it cannot serve as a convenient definition3. As the above discussion suggests, a better-quasi-order is going to be a qo 푄, with no bad sequence, with no bad sequence of sequences, no bad sequence of sequences of sequences, so on and so forth, into the transfinite. To do so we need a convenient notion of ‘index set’ for a sequence of sequences of … of sequences, in short a super-sequence. We now turn to the study of this fundamental notion defined by Nash-Williams.

3The reader who remains unconvinced can try to prove that the partial order (2, =) satisfies this property.

27 2 Sequences in sets and orders

2.2.2 Super-sequences Let us first introduce some useful notations. Given an infinite subset 푋 of 휔 and a natural number 푘, we denote by [푋]푘 the set of subsets of 푋 of cardinality 푘, and by [푋]<∞ the set ⋃ [푋]푘 of finite subsets of 푋. When we 푘∈휔 푘 write an element 푠 ∈ [푋] as {푛0, … , 푛푘−1} we always assume it is written in increasing order 푛0 < 푛1 < … < 푛푘−1 for the usual order on 휔. The cardinality of 푠 ∈ [휔]<∞ is denoted by |푠|. We write [푋]∞ for the set of infinite subsets of 푋. For any 푋 ∈ [휔]∞ and any 푠 ∈ [휔]<∞, we let 푋/푠 = {푘 ∈ 푋 ∣ max 푠 < 푘} and we write 푋/푛 for 푋/{푛}, as we have already done.

Index sets for super-sequences Intuitively super-sequences are sequences of sequences … of sequences. In order to deal properly with this idea we need a convenient notion of index sets. Those will be families of finite sets of natural numbers called fronts. They were defined by Nash-WilliamsNas65 [ ]. As the presence of an ellipsis in the expression ‘sequences of sequences of … of sequences’ suggests, the notion of front admits an inductive definition. To formulate such a definition it is useful to identify the degenerate case of a super-sequence, the level zero of the notion of sequence of … of sequences, namely a function 푓 ∶ 1 → 퐸 which singles out a point of a set 퐸. The index set for these degenerate sequences is the family {∅} called the trivial front. New fronts are then built up from old ones using the following operation.

Definition 2.18. If 푋 ∈ [휔]∞ and 퐹 (푛) ⊆ [푋/푛]<∞ for every 푛 ∈ 푋, we let

seq 퐹 (푛) = {{푛} ∪ 푠 ∣ 푛 ∈ 푋 and 푠 ∈ 퐹 (푛)}. 푛∈푋

Definition 2.19 (Front, inductive definition). A family 퐹 for 푋 ∈ [휔]∞ is a front on 푋 if it belongs to the smallest set of families of finite sets of natural numbers such that

(1) for all 푋 ∈ [휔]∞, the family {∅} is a front on 푋,

(2) if 푋 ∈ [휔]∞ and if 퐹 (푛) is a front on 푋/푛 for all 푛 ∈ 푋, then

퐹 = seq 퐹 (푛) 푛∈푋 is a front on 푋.

28 2 Sequences in sets and orders

Remark 2.20. In the literature, fronts are sometimes called blocks or thin blocks. Since in Section 3.2 and Chapter 3 we have another use for the term block, we follow the terminology of Todorčević [Tod10]. Examples 2.21. We have already seen example of fronts. Indeed for every 푋 ∈ [휔]∞ and every 푛 ∈ 휔 the family [푋]푛 is a front on 푋, where [푋]0 = {∅} is the trivial front. For a new example, consider for every 푛 ∈ 휔 the front [휔/푛]푛 and build 풮 = seq [휔/푛]푛 = {푠 ∈ [휔]<∞ ∣ 1 + min 푠 = |푠|}. 푛∈휔 The front 풮 is traditionally called the Schreier barrier.

⋯ ⋯

1 trivial front front [휔]

⋯ ⋯

2 front [휔] Schreier barrier 풮

Figure 2.2: Pictures of fronts

We defined fronts to make the following: Definition 2.22. A super-sequence in a set 퐸 is a map 푓 ∶ 퐹 → 퐸 from a front into 퐸. Notice that if 퐹 is a non trivial front on 푋, we can recover the unique sequence 퐹 (푛), 푛 ∈ 푋, of fronts from which it is constructed.

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Definition 2.23. For any family 퐹 ⊆ [휔]<∞ and 푛 ∈ 휔 we define the ray of 퐹 at 푛 to be the family

<∞ 퐹푛 = {푠 ∈ [휔/푛] ∣ {푛} ∪ 푠 ∈ 퐹 }.

Then every non trivial front 퐹 on 푋 is built up from its rays 퐹푛, 푛 ∈ 푋, in the sense that: 퐹 = seq 퐹푛. 푛∈푋 Notice that, according to our definition, the trivial front {∅} is a front on 푋 for every 푋 ∈ [휔]∞. Except for this degenerate example, if a family 퐹 ⊆ [푋]<∞ is a front on 푋, then necessarily 푋 is equal to ⋃ 퐹 , the set-theoretic union of the family 퐹 . For this reason we will sometimes say that 퐹 is a front, without reference to any infinite subset 푋 of 휔. Moreover when 퐹 is not trivial, we refer to the unique 푋 for which 퐹 is a front on 푋, namely ⋃ 퐹 , as the base of 퐹 . Importantly, the notion of a front also admits an explicit definition to which we now turn. It makes essential use of the following binary relation.

Definition 2.24. For subsets 푢, 푣 of 휔, we write 푢 ⊑ 푣 when 푢 is an initial segment of 푣, i.e. when 푢 = 푣 or when there exists 푛 ∈ 푣 such that 푢 = {푘 ∈ 푣 ∣ 푘 < 푛}. As usual, we write 푢 ⊏ 푣 for 푢 ⊑ 푣 and 푢 ≠ 푣.

Definition 2.25 (Front, explicit definition). A family 퐹 ⊆ [휔]<∞ is a front on 푋 ∈ [휔]∞ if

(1) either 퐹 = {∅}, or ⋃ 퐹 = 푋,

(2) for all 푠, 푡 ∈ 퐹 푠 ⊑ 푡 implies 푠 = 푡,

(3) (Density) for all 푋′ ∈ [푋]∞ there is an 푠 ∈ 퐹 such that 푠 ⊏ 푋′.

Merely for the purpose of showing that our two definitions coincide, and only until this is achieved, let us refer to a front according to the explicit definition as a fronte. Notice that the family {∅} is a fronte, the trivial fronte. Notice also that if 퐹 is a non trivial fronte then necessarily ∅ ∉ 퐹 . Our first step towards proving the equivalence of our two definitions offronts is the following easy observation.

Lemma 2.26. Let 퐹 be a non trivial fronte on 푋 ∈ [휔]∞. Then for every e 푛 ∈ 푋, the ray 퐹푛 is a front on 푋/푛. Moreover 퐹 = seq푛∈푋 퐹푛.

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Proof. Let 푛 ∈ 푋. For every 푌 ∈ [푋/푛]∞ there exists 푠 ∈ 퐹 with 푠 ⊏ {푛}∪푌 . ′ Since 퐹 is non trivial, 푠 ≠ ∅ and so 푛 ∈ 푠. Therefore 푠 = 푠 ∖ {푛} ∈ 퐹푛 with ′ 푠 ⊏ 푌 , and 퐹푛 satisfies (3). Now if 퐹푛 is not trivial and 푘 ∈ 푋/푛, there is 푠 ∈ 퐹푛 with 푠 ⊏ {푘}∪푋/푘 and necessarily 푘 ∈ 푠 ⊆ ⋃ 퐹푛. Hence ⋃ 퐹푛 = 푋/푛, ′ so condition (1) is met. To see (2), let 푠, 푡 ∈ 퐹푛 with 푠 ⊑ 푡. Then for 푠 = {푛}∪푠 and 푡′ = {푛}∪푡 we have 푠′, 푡′ ∈ 퐹 and 푠′ ⊑ 푡′, so 푠′ = 푡′ and 푠 = 푡, as desired. The last statement is obvious. Our next step consists in assigning a rank to every fronte. To do so, we first recall some classical notions about sequences and trees.

Notations 2.27. For a non empty set 퐴, we write 퐴푛 for the set of sequences 푠 ∶ 푛 → 퐴. Let 퐴<휔 be the set ⋃ 퐴푛 of finite sequences in 퐴. We write 퐴휔 푛∈휔 for the set of infinite sequences 푥 ∶ 휔 → 퐴 in 퐴. Let 푢 ∈ 퐴<휔, 푥 ∈ 퐴<휔 ∪ 퐴휔.

(1) |푥| ∈ 휔 + 1 denotes the length of 푥.

(2) For 푛 ⩽ |푥|, 푥↾푛 is the initial segment, or prefix, of 푥 of length 푛.

(3) We write 푢 ⊑ 푥 if there exists 푛 ⩽ |푥| with 푢 = 푥↾푛. We write 푢 ⊏ 푥 if 푢 ⊑ 푥 and 푢 ≠ 푥.

(4) We write 푢 ⌢ 푥 for the concatenation operation.

Identifying any finite subset of 휔 with its increasing enumeration with respect to the usual order on 휔, we view any fronte as a subset of 휔<휔. Notice that under this identification, our previous definition of ⊑ for subsets of 휔 coincides with the one for sequences.

Definitions 2.28. (1) A tree 푇 on a set 퐴 is a subset of 퐴<휔 that is closed under prefixes, i.e. 푢 ⊑ 푣 and 푣 ∈ 푇 implies 푢 ∈ 푇 .

(2) A tree 푇 on 퐴 is called well-founded if 푇 has no infinite branch, i.e. 휔 if there is no infinite sequence 푥 ∈ 퐴 such that 푥↾푛 ∈ 푇 holds for all 푛 ∈ 휔. In other words, a tree 푇 is well-founded if (푇 , ⊒) is a well-founded partial order.

(3) When 푇 is a non-empty well-founded tree we can define a strictly decreasing function 휌푇 from 푇 to the ordinals by transfinite recursion on the well-founded relation ⊐:

휌푇 (푡) = sup{휌푇 (푠) + 1 ∣ 푡 ⊏ 푠 ∈ 푇 } for all 푡 ∈ 푇 .

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It is easily shown to be equivalent to

⌢ ⌢ 휌푇 (푡) = sup{휌푇 (푡 (푎)) + 1 ∣ 푎 ∈ 퐴 and 푡 (푎) ∈ 푇 } for all 푡 ∈ 푇 .

The rank of the non-empty well-founded tree 푇 is the ordinal 휌푇 (∅). For any fronte 퐹 , we let 푇 (퐹 ) be the smallest tree on 휔 containing 퐹 , i.e.

푇 (퐹 ) = {푠 ∈ 휔<휔 ∣ ∃푡 ∈ 퐹 푠 ⊑ 푡}.

The following is a direct consequence of the explicit definition of a front.

Lemma 2.29. For every fronte 퐹 , the tree 푇 (퐹 ) is well-founded.

Proof. If 푥 is an infinite branch of 푇 (퐹 ), then 푥 enumerates an infinite subset 푋 of ⋃ 퐹 such that for every 푢 ⊏ 푋 there exists 푡 ∈ 퐹 with 푢 ⊑ 푡. Since 퐹 is a fronte there exists a (unique) 푠 ∈ 퐹 with 푠 ⊏ 푋. But for 푛 = min 푋/푠, 푢 = 푠 ∪ {푛} and there is 푡 ∈ 퐹 with 푢 ⊑ 푡. But then 퐹 ∋ 푠 ⊏ 푢 ⊑ 푡 ∈ 퐹 contradicting the explicit definition of a front.

Definition 2.30. Let 퐹 be a fronte. The rank of 퐹 , denoted by rk 퐹 , is the rank of the tree 푇 (퐹 ).

Example 2.31. Notice that the family {∅} is the only fronte of null rank, and for all positive integer 푛, the front [휔]푛 has rank 푛. Moreover the Schreier barrier 풮 has rank 휔. We now observe that the rank of 퐹 is closely related to the rank of its rays e ∞ 퐹푛, 푛 ∈ 푋. Let 퐹 be a non trivial front on 푋 ∈ [휔] and recall that by e Lemma 2.26, the ray 퐹푛 is a front on 푋/푛 for every 푛 ∈ 푋. Now notice that e the tree 푇 (퐹푛) of the front 퐹푛 is naturally isomorphic to the subset {푠 ∈ 푇 (퐹 ) ∣ {푛} ⊑ 푠} of 푇 (퐹 ). The rank of the fronte 퐹 is therefore related to the ranks of its rays through the following formula:

rk 퐹 = sup{rk(퐹푛) + 1 ∣ 푛 ∈ 푋}.

In particular, rk 퐹푛 < rk 퐹 for all 푛 ∈ 푋. This simple remark allows one to prove results on frontes by induction on the rank by applying the induction hypothesis to the rays, as it was first done by Pudlák and Rödl [PR82]. It also allows us to prove that the two definitions of a front that we gave actually coincide.

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Lemma 2.32. The explicit definition and the inductive definition of afront coincide. Proof. Inductive → Explicit: The family {∅} is the trivial fronte. Now let 푋 ∈ ∞ e [휔] and suppose that 퐹푛 is a front on 푋/푛 for all 푛 ∈ 푋. We need to see e that 퐹 = seq푛∈푋 퐹푛 is a front on 푋. Clearly ⋃ 퐹 = 푋. If 푠, 푡 ∈ 퐹 and 푠 ⊑ 푡, then min 푠 = min 푡 and so for 푠′ = 푠∖{min 푠} and 푡′ = 푡∖{min 푡} ′ ′ ′ ′ ′ ′ we have 푠 , 푡 ∈ 퐹푛 and 푠 ⊑ 푡 , hence 푠 = 푡 and 푠 = 푡. Finally, if ∞ ′ ′ 푌 ∈ [푋] with 푛 = min 푌 , then there exists 푠 ∈ 퐹푛 with 푠 ⊏ 푌 ∖ {푛} and so 푠 = {푛} ∪ 푠′ ∈ 퐹 and 푠 ⊏ 푌 . So 퐹 is a fronte, as desired.

Explicit → Inductive: We show that every fronte 퐹 satisfies the inductive definition of a front by induction on the rankof 퐹 . If rk 퐹 = 0, then 퐹 = {∅} is a front according to the inductive definition. Now suppose 퐹 is a front according to the explicit definition with rk 퐹 > 0. In particular ∞ e ⋃ 퐹 = 푋 for some 푋 ∈ [휔] . By Lemma 2.26, 퐹푛 is a front for every 푛 ∈ 푋, and as rk 퐹푛 < rk 퐹 for every 푛 ∈ 푋, it follows that 퐹푛 is a front according to the inductive definition, by induction hypothesis. Finally

as 퐹 = seq푛∈푋 퐹푛, we get that 퐹 is a front according to the inductive definition. Finally notice that the rank of a front naturally arise from the inductive definition. Let 픉0 be the set containing only the trivial front. Then for any countable ordinal 훼, let 퐹 ∈ 픉 if 퐹 ∈ ⋃ 픉 or 퐹 = seq 퐹 where 훼 훽<훼 훽 푛∈푋 푛 푋 ∈ [휔]∞ and each 퐹 is a front on 푋/푛 which belongs to some 픉 for some 푛 훽푛 훽푛 < 훼. Then clearly the set of all fronts is equal ⋃ 픉훼. Now it should be 훼<휔1 clear that for every front 퐹 the smallest 훼 < 휔1 for which 퐹 ∈ 픉훼 is rk 퐹 , the rank of 퐹 .

Sub-front and sub-super-sequences When dealing with arbitrary super-sequences we will be particularly interested in extracting sub-super-sequences which enjoy further properties. Definition 2.33. A sub-super-sequence of a super-sequence 푓 ∶ 퐹 → 퐸 is a restriction 푓↾퐺 ∶ 퐺 → 퐸 to some front 퐺 included in 퐹 . The following important operation allows us to understand the sub-fronts of a given front, i.e. sub-families of a front which are themselves fronts. For a family 퐹 ⊆ 풫(휔) and some 푋 ∈ [휔]∞, we define the sub-family

퐹 |푋 ∶= {푠 ∈ 퐹 ∣ 푠 ⊆ 푋}.

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Proposition 2.34. Let 퐹 be a front on 푋. Then a family 퐹 ′ ⊆ 퐹 is a front if and only if there exists 푌 ∈ [푋]∞ such that 퐹 |푌 = 퐹 ′.

Proof. The claim is obvious if 퐹 is trivial so suppose 퐹 is non-trivial.

→ Let 퐹 ′ ⊆ 퐹 be a front on 푌 . Since 퐹 ′ is not trivial either, 푌 = ⋃ 퐹 ′ ⊆ ⋃ 퐹 = 푋. Now if 푠 ∈ 퐹 ′ then clearly 푠 ∈ 퐹 |푌 . Conversely if 푠 ∈ 퐹 |푌 then there exists a unique 푡 ∈ 퐹 ′ with 푡 ⊏ 푠 ∪ 푌 /푠 and so either 푠 ⊑ 푡 or 푡 ⊑ 푠. Since 퐹 is a front and 푠, 푡 ∈ 퐹 , necessarily 푠 = 푡 and so 푠 ∈ 퐹 ′. Therefore 퐹 ′ = 퐹 |푌 .

← If 푌 ∈ [푋]∞ then the family 퐹 |푌 is a front on 푌 . Clearly 퐹 |푌 satisfies (2). If 푍 ∈ [푌 ]∞ then since 푌 ⊆ 푋, then 푍 ∈ [푋]∞ and so there exists 푠 ∈ 퐹 with 푠 ⊏ 푍. But then 푠 ⊆ 푍 ⊆ 푌 , so in fact 푠 ∈ 퐹 |푌 and therefore 퐹 |푌 satisfies (3). For (1), notice that ⋃ 퐹 |푌 ⊆ 푌 by definition and that if 푛 ∈ 푌 , then as we have already seen there exists 푠 ∈ 퐹 |푌 with 푠 ⊏ {푛} ∪ 푌 /푛, so 푛 ∈ 푠 and 푛 ∈ ⋃ 퐹 |푌 .

Observe that the operation of restriction commutes with the taking of rays.

Fact 2.35. Let 퐹 ⊆ 풫(휔) and 푋 ∈ [휔]∞. For every 푛 ∈ 푋 we have

퐹푛|푋 = (퐹 |푋)푛.

Notice also the following simple important fact. If 퐹 ′ is a sub-front of a front 퐹 , then the tree 푇 (퐹 ′) is included in the tree 푇 (퐹 ) and so rk 퐹 ′ ⩽ rk 퐹 . Throughout this thesis we extensively make use of the following fundamental theorem by Nash-Williams: Any time we partition a front into finitely many pieces, at least one of the pieces must contain a front.

Theorem 2.36 (Nash-Williams). Let 퐹 be a front. For any subset 푆 of 퐹 there exists a front 퐹 ′ ⊆ 퐹 such that either 퐹 ′ ⊆ 푆 or 퐹 ′ ∩ 푆 = ∅.

We now prove this theorem to give a simple example of a proof by induction on the rank of a front, a technique we make use of on several occasions in Chapter 3. Proof. The claim is obvious for the trivial front whose only subsets are the empty set and the whole trivial front. So suppose that the claim holds for every front of rank smaller than 훼. Let 퐹 be a front on 푋 with rk 퐹 = 훼 and 푆 ⊆ 퐹 . For every 푛 ∈ 푋 let 푆푛 be the subset of the ray 퐹푛 given by 푆푛 = {푠 ∈ 퐹푛 ∣ {푛} ∪ 푠 ∈ 푆}.

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Set 푋 = 푋 and 푛 = min 푋 . Since rk 퐹 < 훼 there exists by induction −1 0 −1 푛0 ∞ hypothesis some 푋0 ∈ [푋−1/푛0] such that either 퐹 |푋 ⊆ 푆 , or 퐹 |푋 ∩ 푆 = ∅. 푛0 0 푛0 푛0 0 푛0 Set 푛 = min 푋 . Now applying the induction hypothesis to 퐹 |(푋 /푛 ) and 1 0 푛1 0 0 푆 we get an 푋 ∈ [푋 /푛 ]∞ such that either 퐹 |푋 ⊆ 푆 , or 퐹 |푋 ∩ 푛1 1 0 0 푛1 1 푛1 푛1 1 푆 = ∅. Continuing in this fashion, we obtain a sequence 푋 together with 푛1 푘 ∞ 푛푘 = min 푋푘−1 such that for all 푘 we have 푋푘 ∈ [푋푘−1/푛푘] and either 퐹 |푋 ⊆ 푆 , or 퐹 |푋 ∩ 푆 = ∅. 푛푘 푘 푛푘 푛푘 푘 푛푘 Now there exists 푌 ∈ [휔]∞ such that either 퐹 |푋 ⊆ 푆 for all 푘 ∈ 푌 , 푛푘 푘 푛푘 or 퐹 |푋 ∩ 푆 = ∅ for all 푘 ∈ 푌 . Let 푋 = {푛 ∣ 푘 ∈ 푌 } then 퐹 |푋 is 푛푘 푘 푛푘 푘 as desired. Indeed for all 푠 ∈ 퐹 |푋 we have min 푠 = 푛푘 for some 푘 ∈ 푌 and 푠∖{푛 } ∈ 퐹 |푋 . Hence by the choice of 푌 , either 푠∖{min 푠} ∈ 푆 for all 푘 푛푘 푘 min 푠 푠 ∈ 퐹 |푋, or 푠 ∖ {min 푠} ∉ 푆min 푠 for all 푠 ∈ 퐹 |푋. Therefore either 퐹 |푋 ⊆ 푆 or 퐹 |푋 ∩ 푆 = ∅.

2.2.3 Multi-sequences Another approach to super-sequences initiated by Simpson [Sim85] has proved very useful in the theory of better-quasi-orders. We now describe this approach and relate it to super-sequences. Let 퐸 be any set, and 푓 ∶ 퐹 → 퐸 be a super-sequence with 퐹 a front on 푋. By the explicit definition of front for every 푌 ∈ [푋]∞ there exists a unique 푠 ∈ 퐹 with 푠 ⊏ 푌 . We can therefore define a map 푓↑ ∶ [푋]∞ → 퐸 defined by 푓↑(푌 ) = 푓(푠) where 푠 is the unique member of 퐹 with 푠 ⊏ 푌 . Definition 2.37. A multi-sequence into some set 퐸 is a map ℎ ∶ [푋]∞ → 퐸 for some 푋 ∈ [휔]∞.A sub-multi-sequence of ℎ ∶ [푋]∞ → 퐸 is a restriction of ℎ to [푌 ]∞ for some 푌 ∈ [푋]∞. For every 푋 ∈ [휔]∞ we endow [푋]∞ with the topology induced by the Cantor space, viewing subsets as their characteristic functions. As a topological space [푋]∞ is homeomorphic to the Baire space 휔휔. This homeomorphism is conveniently realised via the embedding of [푋]∞ into 휔휔 which maps each 푌 ∈ ∞ [푋] to its injective and increasing enumeration 푒푌 ∶ 휔 → 푌 . We henceforth identify the space [푋]∞ with the closed subset of 휔휔 of injective and increasing sequences in 푋. From this point of view we have a countable basis of clopen sets for [푋]∞ consisting in sets of the form

∞ ∞ <∞ 푀푠 = 푁푠 ∩ [푋] = {푌 ∈ [푋] ∣ 푠 ⊏ 푌 }, for 푠 ∈ [푋] .

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Definition 2.38. A multi-sequence ℎ ∶ [푋]∞ → 퐸 is locally constant if for ∞ <∞ all 푌 ∈ [푋] there exists 푠 ∈ [푋] such that 푌 ∈ 푀푠 and ℎ is constant on ∞ ∞ 푀푠, i.e. for every 푌 ∈ [푋] there exists 푠 ⊏ 푌 such that for every 푍 ∈ [푋] , 푠 ⊏ 푍 implies ℎ(푍) = ℎ(푌 ). Clearly for every super-sequence 푓 ∶ 퐹 → 퐸 where 퐹 is a front on 푋 the map 푓↑ ∶ [푋]∞ → 퐸 is locally constant. Conversely for any locally constant multi-sequence ℎ ∶ [푋]∞ → 퐸, let

ℎ <∞ 푆 = {푠 ∈ [푋] ∣ ℎ is constant on 푀푠}. Lemma 2.39. The set 퐹 ℎ of ⊑-minimal elements of 푆ℎ is a front on 푋. Proof. By ⊑-minimality if 푠, 푡 ∈ 퐹 ℎ and 푠 ⊑ 푡, then 푠 = 푡. For every 푌 ∈ [푋]∞, since ℎ is locally constant there exists 푠 ⊏ 푌 such that ℎ is constant on 푀푠. Hence there exists 푡 ∈ 퐹 ℎ with 푡 ⊑ 푠, and so 푡 ⊏ 푌 too. To see that either 퐹 ℎ is trivial or ⋃ 퐹 ℎ = 푋, notice that ℎ is constant if and only if 퐹 ℎ is the ℎ trivial front if and only if ∅ ∈ 퐹ℎ. So if 퐹 is not trivial, then for every 푛 ∈ 푋 there exists 푠 ∈ 퐹 ℎ with 푠 ⊏ {푛} ∪ 푋/푛 and since 푠 ≠ ∅, we get 푛 ∈ 푠 and 푛 ∈ ⋃ 퐹 ℎ. We can therefore associate to every locally constant multi-sequence ℎ ∶ [푋]∞ → 퐸 a super-sequence ℎ↓ ∶ 퐹 ℎ → 퐸 by letting, in the obvious way, ↓ ℎ ℎ (푠) be equal to the unique value taken by ℎ on 푀푠 for every 푠 ∈ 퐹 . Remark 2.40. Clearly every front arises as an 퐹 ℎ for some locally constant multi-sequence ℎ. Indeed for any front 퐹 and any injective super-sequence 푓 from 퐹 , we have 퐹 = 퐹 푓↑ . Therefore we can think of the definition of a front as a characterisation of those families of finite subsets of 휔 arising as an 퐹 ℎ for some locally constant multi-sequence ℎ. The basic properties of the correspondence ℎ ↦ ℎ↓ and 푓 ↦ 푓 ↑ are easily stated with the help of the following partial order among super-sequences into a given set. Definition 2.41. Let both 퐹 and 퐺 be fronts on the same set 푋 ∈ [휔]∞ and 푓 ∶ 퐹 → 퐸 and 푔 ∶ 퐺 → 퐸 be any maps. We write 푓 ⊑ 푔 if 퐹 ⊆ 퐺 and for every 푠 ∈ 퐹 and every 푡 ∈ 퐺, 푠 ⊑ 푡 implies 푓(푠) = 푔(푡). To simplify notation we write 푓̌ ∶ 퐹̌ → 퐸 instead of (푓↑)↓ ∶ 퐹 푓↑ → 퐸. Fact 2.42. Let 푋 ∈ [휔]∞ and 퐸 be a set. (i) for every front 퐹 on 푋 and every map 푓 ∶ 퐹 → 퐸, the map 푓̌ ∶ 퐹̌ → 퐸 is such that 푓̌ ⊑ 푓.

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(ii) for every fronts 퐹 and 퐺 on 푋 and maps 푓 ∶ 퐹 → 퐸 and 푔 ∶ 퐺 → 퐸, 푓 ⊑ 푔 implies 푓↑ = 푔↑. (iii) for every locally constant map ℎ ∶ [푋]∞ → 퐸, we have (ℎ↓)↑ = ℎ. It follows that for every locally constant multi-sequence ℎ ∶ [푋]∞ → 퐸 the super-sequence ℎ↓ ∶ 퐹 ℎ → 퐸 is the minimal element for ⊑ among the set of super-sequences 푔 ∶ 퐺 → 퐸 with 푔↑ = ℎ. Moreover for every super-sequence 푓 ∶ 퐹 → 퐸 the super-sequence 푓̌ ∶ 퐹̌ → 퐸 is the ⊑-minimal among the ̌ super-sequences 푔 with 푔 ⊑ 푓. In particular 푓̌ = 푓 ̌ for every super-sequence 푓. The super-sequences which are ⊑-minimal will play a rôle later and deserve a name. Definition 2.43. Let 퐸 be a set and 퐹 a front on 푋. A super-sequence 푓 ∶ 퐹 → 퐸 is said to be spare if 푓 is minimal for ⊑, or equivalently 푓̌ = 푓, i.e. if 퐹̌ = 퐹 . Example 2.44. If 푐 ∶ 퐹 → 퐸 is constant equal to 푒 ∈ 퐸 then 푐 is not spare and of course ̌푐∶ {∅} → 퐸, ∅ ↦ 푒. The following is a simple characterisation of spare super-sequences. Lemma 2.45. Let 푓 ∶ 퐹 → 퐸 be a map from a front to some set 퐸. Then the following are equivalent (i) 푓 is spare, (ii) for every 푠 ∈ 퐹 , if for every 푡, 푡′ ∈ 퐹 푠 ⊑ 푡 and 푠 ⊑ 푡′ imply 푓(푡) = 푓(푡′), then 푠 ∈ 퐹 . Proof. Suppose that 푠 ∈ 퐹 ∖ 퐹 is such that for every 푡, 푡′ ∈ 퐹 , 푠 ⊑ 푡 and ′ ′ ̂ 푓 ̂ 푠 ⊑ 푡 imply 푓(푡) = 푓(푡 ). Then 푓 is constant on 푀푠 but 푠 ⊏ 푡 ∈ 퐹 so 푡 ∉ 퐹 . Therefore 푓 is not spare. Conversely if 푓 is not spare, then there exists 푡 ∈ 퐹 ∖ 퐹̌. This means that ̌ ↑ ′ there is 푠 ∈ 퐹 with 푠 ⊏ 푡 and 푓 is constant on 푀푠, so for every 푡 ∈ 퐹 with 푠 ⊑ 푡′ we have 푓(푡) = 푓(푡′).

2.2.4 Iterated powerset, determinacy of finite games It also transpires that if, by a certain fairly natural extension of our definition of [풫푛(푄)], we define [풫훼(푄)] for every ordinal 훼, then 푄 is bqo iff [풫훼(푄)] is wqo for every ordinal 훼. To justify these statements would not be relevant here, but it was from this point of view that the author was first led to study bqo sets.

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Crispin St. John Alvah Nash-Williams [Nas65, p. 700] Following in Nash-Williams’ steps, we introduce the notion of better-quasi- orders as the quasi-orders whose iterated powersets are wqo. We do this in the light of further developments of the theory, taking advantage of Simpson’s point of view on super-sequences, using the determinacy of finite games and a powerful game-theoretic technique invented by Tony Martin. First let us define precisely the iterated powerset of a qo together withits lifted quasi-order. To facilitate the following discussion we focus on the non- empty sets over some quasi-order 푄. Let 풫∗(퐴) denote the set of non-empty subsets of a set 퐴, i.e. 풫∗(퐴) = 풫(퐴) ∖ {∅}. We define by transfinite recursion

∗ 푉0 (푄) = 푄 ∗ ∗ ∗ 푉훼+1(푄) = 풫 (푉훼(푄)) ∗ ∗ 푉휆 (푄) = ⋃ 푉훼(푄), for 휆 limit. 훼<휆 We treat the element of 푄 as urelements or atoms, namely they have no elements but they are different from the empty set. Let

∗ ∗ 푉 (푄) = ⋃ 푉훼(푄). 훼 Let us say that a set 퐴 is 푄-transitive if for all 푥 ∈ 퐴 we have 푥 ⊆ 퐴 whenever 푥∈푄̸ . We define the 푄-transitive closure of 푋 ∈ 푉 ∗(푄), denoted by tcQ(푋), as the smallest 푄-transitive set containing 푋. Notice that in particular tcQ(푞) = {푞} for every 푞 ∈ 푄 and that tcQ(푋) = 푋 for every non-empty subset 푋 of 푄. Finally we define for every 푋 ∈ 푉 ∗(푄) the support of 푋, denoted by suppQ(푋), as the set of elements of 푄 which belong to tcQ(푋):

suppQ(푋) = 푄 ∩ tcQ(푋).

Notice that suppQ(푋) is never empty and that for every 푞 ∈ 푄 we have suppQ(푞) = {푞}. Following an idea of Forster [For03] we define the quasi-order on 푉 ∗(푄) via the existence of a winning strategy in a natural game. We refer the reader to Kechris [Kec95, (20.)] for the basic definitions pertaining to two-player games with perfect information. Definition 2.46. For every 푋, 푌 ∈ 푉 ∗(푄) we define a two-player game with perfect information 퐺푉 ∗ (푋, 푌 ) by induction on the membership relation. The ′ game 퐺푉 ∗ (푋, 푌 ) goes as follows. Player I starts by choosing some 푋 such that:

38 2 Sequences in sets and orders

• if 푋 ∉ 푄, then 푋′ ∈ 푋,

• otherwise, 푋′ = 푋.

Then Player II replies by choosing some 푌 ′ such that:

• if 푌 ∉ 푄, then 푌 ′ ∈ 푌 ,

• otherwise 푌 ′ = 푌 .

If both 푋′ and 푌 ′ belong to 푄, then Player II wins if 푋′ ⩽ 푌 ′ in 푄 and Player ′ ⩽̸ ′ ′ ′ I wins if 푋 푌 . Otherwise the game continues as in 퐺푉 ∗ (푋 , 푌 ). We then define the lifted quasi-order on 푉 ∗(푄) by letting for 푋, 푌 ∈ 푉 ∗(푄)

푋 ⩽ 푌 ⟷ Player II has a winning strategy in 퐺푉 ∗ (푋, 푌 ).

Remark 2.47. The above definition can be rephrased by induction on the membership relation as follows:

(1) if 푋, 푌 ∈ 푄, then 푋 ⩽ 푌 if and only if 푋 ⩽ 푌 in 푄,

(2) if 푋 ∈ 푄 and 푌 ∉ 푄, then

푋 ⩽ 푌 ⟷ there exists 푌 ′ ∈ 푌 with 푋 ⩽ 푌 ′,

(3) if 푋 ∉ 푄 and 푌 ∈ 푄, then

푋 ⩽ 푌 ⟷ for every 푋′ ∈ 푋 we have 푋′ ⩽ 푌 ,

(4) if 푋 ∉ 푄 and 푌 ∉ 푄, then

푋 ⩽ 푌 ⟷ for every 푋′ ∈ 푋 there exists 푌 ′ ∈ 푌 with 푋′ ⩽ 푌 ′.

Our definition coincides with the one given by Shelah [She82, Claim 1.7, p.188]. But Milner [Mil85] and Laver [Lav71] both omit condition (3).

The axiom of foundation ensures that in any play of a game 퐺푉 ∗ (푋, 푌 ) a round where both players have chosen elements of 푄 is eventually reached, resulting in the victory of one of the two players. In particular, each game 퐺푉 ∗ (푋, 푌 ) is determined as already proved by Von Neumann and Morgen- stern [VM44] (see [Kec95, (20.1)]). The crucial advantage of the game-theoretic formulation of the quasi-order on 푉 ∗(푄) resides in the fact that the negative

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퐺 ∗ (푋 , 푋 ) 퐺 ∗ (푋 , 푋 ) 퐺 ∗ (푋 , 푋 ) 퐺 ∗ (푋 , 푋 ) 퐺 ∗ (푋 , 푋 ) 푉 푛0 푛1 푉 푛1 푛2 푉 푛2 푛3 푉 푛3 푛4 푉 푛4 푛5 ⎩ { { { { ⎨ { { { { ⎧ ⎩ { { { { ⎨ { { { { ⎧ ⎩ { { { ⎨ { { { ⎧ ⎩ { { { ⎨ { { { ⎧ ⎩ { { { ⎨ { { { ⎧ 휎 휎 푛 ,푛 푛1,푛2 휎푛 ,푛 휎푛 ,푛 휎 I 0 1 II I II I 2 3II I 3 4II I 푛4,푛5II

0 0 0 0 0 0 0 0 0 푌0 푌1 푌1 푌2 푌2 푌3 푌3 푌4 푌4

1 ⩽̸ 1 1 1 1 1 1 푌0 푌1 푌1 푌2 푌2 푌3 푌3 ∣ ∣ ∣ ∣ ∣ = 2 ⩽̸ 2 2 푌1 푌2 푌2 ∣ ∣ ∣ ∣ ∣ ⩽̸ 푔(푁) 푔(∗푁)

Figure 2.3: Constructing a multi-sequence by stringing strategies together. condition 푋 ⩽̸ 푌 is equivalent to the existential statement ‘Player I has a winning strategy’. ∗ Now suppose 푄 is a quasi-order such that 푉 (푄) is not wqo and let (푋푛)푛∈휔 ∗ ⩽̸ be a bad sequence in 푉 (푄). Whenever 푚 < 푛 we have 푋푚 푋푛 and we can choose a winning strategy 휎푚,푛 for Player I in 퐺푉 ∗ (푋푚, 푋푛). We define a loc- ∞ ally constant multi-sequence 푔 ∶ [휔] → 푄 as follows. Let 푁 = {푛0, 푛1, 푛2, …} be an infinite subset of 휔 enumerated in increasing order. We define 푔(푁) as the last move of Player I in a particular play of 퐺 ∗ (푋 , 푋 ) in a way best 푉 푛0 푛1 understood by contemplating Figure 2.3. 0 Let 푌 be the the first move of Player I in 퐺 ∗ (푋 , 푋 ) as prescribed 0 푉 푛0 푛1 by its winning strategy 휎 . Then let Player II copy the first move 푌 0 푛0,푛1 1 of Player I given by the strategy 휎 in 퐺 ∗ (푋 , 푋 ). Then Player I 푛1,푛2 푉 푛1 푛2 answers 푌 1 according to the strategy 휎 . Now if 푌 0 is not in 푄, then we 0 푛0,푛1 1 need to continue our play of 퐺 ∗ (푋 , 푋 ) a little further to determine the 푉 푛1 푛2 second move of Player II in 퐺 ∗ (푋 , 푋 ). Let the first move of Player II in 푉 푛0 푛1 퐺 ∗ (푋 , 푋 ) be the first move of Player I in 퐺 ∗ (푋 , 푋 ) as prescribed 푉 푛1 푛2 푉 푛2 푛3 by his winning strategy 휎 . Then this determines the second move 푌 1 of 푛2,푛3 1 Player I in 퐺 ∗ (푋 , 푋 ) according to 휎 . We then let the second move 푉 푛1 푛2 푛1,푛2 1 1 of Player II in 퐺 ∗ (푋 , 푋 ) to be this 푌 . This yields some answer 푌 of 푉 푛0 푛1 1 0 Player I according to 휎 . We continue so on and so forth until the play 푛0,푛1 푘푁 푘푁 of 퐺 ∗ (푋 , 푋 ) reaches an end with some (푌 , 푌 ) ∈ 푄 × 푄 and we let 푉 푛0 푛1 0 1 푘푁 푔(푁) = 푌 . Since the play of 퐺 ∗ (푋 , 푋 ) is finite, 푔(푁) depends only on 0 푉 푛0 푛1

40 2 Sequences in sets and orders a finite initial segment of 푁 and we have therefore defined a locally constant multi-sequence 푔 ∶ [휔]∞ → 푄. Now since Player I has followed the winning strategy 휎 we have 푌 푘푁 ⩽̸ 푛0,푛1 0 푘푁 푌 . Now if the play of the game 퐺 ∗ (푋 , 푋 ) has not yet reached an end 1 푉 푛1 푛2 at step 푘푁 we go on in the same fashion. Assume it ends with some pair 푙 푙 푘푁 (푌1 , 푌2 ) in 푄. By the rules of the game 퐺푉 ∗ , since 푌1 ∈ 푄 we necessarily 푙 푘푁 푙 ∞ have 푌1 = 푌1 . But 푌1 is just 푔({푛1, 푛2, 푛3, …}), hence for every 푁 ∈ [휔] we have 푔(푁) ⩽̸ 푔(푁 ∖ {min 푁}). ∞ For every 푁 ∈ [휔] we call the shift of 푁, denoted by ∗푁, the set 푁 ∖ {min 푁}. We are led to the following:

Definition 2.48. Let 푄 be a qo and ℎ ∶ [푋]∞ → 푄 a multi-sequence. ⩽̸ ∞ (1) We say that ℎ is bad if ℎ(푁) ℎ(∗푁) for every 푁 ∈ [푋] ,

∞ (2) We say that ℎ is good if there exists 푁 ∈ [푋] with ℎ(푁) ⩽ ℎ(∗푁), At last, we present the deep definition due to Nash-Williams here in a modern reformulation.

Definition 2.49. A quasi-order 푄 is a better-quasi-order (bqo) if there is no bad locally constant multi-sequence into 푄.

Of course the definition of better-quasi-order can be formulated in terms of super-sequences as Nash-Williams originally did. The only missing ingredient is a counterpart of the shift map 푁 ↦ ∗푁 on finite subsets of natural numbers. Definition 2.50. For 푠, 푡 ∈ [휔]<∞ we say that 푡 is a shift of 푠 and write 푠 ¡ 푡 if there exists 푋 ∈ [휔]∞ such that

푠 ⊏ 푋 and 푡 ⊏ ∗푋. Definitions 2.51. Let 푄 be a qo and 푓 ∶ 퐹 → 푄 be a super-sequence.

(1) We say that 푓 is bad if whenever 푠 ¡ 푡 in 퐹 , we have 푓(푠) ⩽̸ 푓(푡).

(2) We say that 푓 is good if there exists 푠, 푡 ∈ 퐹 with 푠 ¡ 푡 and 푓(푠) ⩽ 푓(푡).

Lemma 2.52. Let 푄 be a quasi-order.

(i) If ℎ ∶ [휔]∞ → 푄 is locally constant and bad, then ℎ↓ ∶ 퐹 ℎ → 푄 is a bad super-sequence.

41 2 Sequences in sets and orders

(ii) If 푓 ∶ 퐹 → 푄 is a bad super-sequence from a front on 푋, then 푓↑ ∶ [푋]∞ → 푄 is a bad locally constant multi-sequence.

Proof. (i) Suppose ℎ ∶ [푋]∞ → 푄 is locally constant and bad. Let us show that ℎ↓ ∶ 퐹 ℎ → 푄 is bad. If 푠, 푡 ∈ 퐹 ℎ with 푠 ¡ 푡, i.e. there exists ∞ ↓ 푌 ∈ [푋] such that 푠 ⊏ 푌 and 푡 ⊏ ∗푌 . Then ℎ (푠) = ℎ(푌 ) and ↓ ↓ ⩽̸ ↓ ℎ (푡) = ℎ(∗푌 ) and since ℎ is assumed to be bad, we have ℎ (푠) ℎ (푡). (ii) Suppose 푓 ∶ 퐹 → 푄 is bad from a front on 푋 and let 푌 ∈ [푋]∞. There are ↑ unique 푠, 푡 ∈ 퐹 such that 푠 ⊏ 푌 and 푡 ⊏ ∗푌 , and clearly 푓 (푌 ) = 푓(푠), ↑ ¡ ↑ ⩽̸ ↑ 푓 (∗푌 ) = 푓(푡), and 푠 푡. Therefore 푓 (푋) 푓 (∗푋) holds. Proposition 2.53. For a quasi-order 푄 the following are equivalent.

(i) 푄 is a better-quasi-order,

(ii) there is no bad super-sequence into 푄,

(iii) there is no bad spare super-sequence into 푄.

The idea of stringing strategies together that we used to arrive at the definition of bqo is directly inspired from a famous technique used by van Engelen, Miller, and Steel [vEMS87, Theorem 3.2] together with Louveau and Saint Raymond [LS90, Theorem 3]. This method was first applied by Martin in the proof of the well-foundedness of the Wadge hierarchy (see [Kec95, (21.15), p. 158]). Forster [For03] introduces better-quasi-orders in a very similar way, but a super-sequence instead of a multi-sequence is constructed, making the sim- ilarity with the method used by van Engelen, Miller, and Steel [vEMS87] and Louveau and Saint Raymond [LS90] less obvious. One of the advantages of multi-sequences resides in the fact that they enable us to work with super- sequences without explicitly referring to their domains. This is particularly useful in the above construction, since a bad sequence in 푉 ∗(푄) can yield a multi-sequence whose underlying front is of arbitrarily large rank. Indeed Mar- cone [Mar94] showed that super-sequences from fronts of arbitrarily large rank are required in the definition of bqo. Notice that the notion of bqo naturally lies between those of well-orders and wqo.

Proposition 2.54. Let 푄 be a qo. Then

푄 is a well-order → 푄 is bqo → 푄 is wqo.

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Proof. Suppose 푄 is a well order and let ℎ ∶ [휔]∞ → 푄 is any multi-sequence ∞ into 푄. Fix 푋 ∈ [휔] and let 푋0 = 푋 and 푋푛+1 = ∗푋푛. Since 푄 is a well- order, there exists 푛 such that ℎ(푋푛) ⩽ ℎ(푥푛+1), otherwise ℎ(푋푛) would be a descending chain in 푄. So ℎ is good and therefore 푄 is bqo. Now observe that for 푚, 푛 ∈ {휔} we have {푚} ¡ {푛} if and only if 푚 < 푛. So if 푄 is bqo, then in particular every sequence 푓 ∶ [휔]1 → 푄 is good, and so 푄 is wqo.

2.2.5 Equivalence Pushing further the idea that led us to the definition of bqo, we can build from any bad multi-sequence in 푉 ∗(푄) a bad multi-sequence in 푄. Therefore proving that if 푄 is bqo, then 푉 ∗(푄) is actually bqo.

Proposition 2.55. Let 푄 be a qo. For every bad locally constant ℎ ∶ [휔]∞ → 푉 ∗(푄) there exists a bad locally constant 푔 ∶ [휔]∞ → 푄 such that 푔(푋) ∈ ∞ suppQ(ℎ(푋)) for every 푋 ∈ [휔] . Proof. Let ℎ ∶ [휔]∞ → 푉 ∗(푄) be locally constant and bad, and let us write ∞ ℎ(푋) = ℎ푋 for 푋 ∈ [휔] . Notice that the image of ℎ is countable and choose ∞ for every 푋 ∈ [휔] a winning strategy 휎 for Player I in 퐺 ∗ (ℎ , ℎ ). We 푋 푉 푋 ∗푋 let 푋0 = 푋 and 푋푛+1 = ∗푋푛.

퐺 ∗ (ℎ , ℎ ) 퐺 ∗ (ℎ , ℎ ) 퐺 ∗ (ℎ , ℎ ) 퐺 ∗ (ℎ , ℎ ) 퐺 ∗ (ℎ , ℎ )퐺 ∗ (ℎ , ℎ ) 푉 푋0 푋1 푉 푋1 푋2 푉 푋2 푋3 푉 푋3 푋4 푉 푋4 푋5 푉 푋5 푋6 ⎩ { { ⎨ { { ⎧ ⎩ { { ⎨ { { ⎧ ⎩ { { ⎨ { { ⎧ ⎩ { { ⎨ { { ⎧ ⎩ { { ⎨ { { ⎧ ⎩ { { ⎨ { { ⎧ 휎 휎 휎 휎 휎 휎 I 0 II I 1 II I 2 II I 3 II I 4 II I 5 II

0 휎0 0 0 휎1 0 0 휎2 0 0 휎3 0 0 휎4 0 0 푌0 푌1 푌1 푌2 푌2 푌3 푌3 푌4 푌4 푌5 푌5

1 휎0 1 1 휎1 1 1 휎2 1 1 휎3 1 1 푌0 푌1 푌1 푌2 푌2 푌3 푌3 푌4 푌4

2 ⩽̸ 2 2 휎1 2 2 휎2 2 2 푌0 푌1 푌1 푌2 푌2 푌3 푌3 ∣ ∣ ∣ ∣ ∣ = 3 ⩽̸ 3 3 푌1 푌2 푌2 ∣ ∣ ∣ ∣ ∣

Figure 2.4: Stringing strategies together.

Consider the diagram in Figure 2.4 obtained by letting Player I follow the winning strategy 휎 = 휎 in 퐺 ∗ (ℎ , ℎ ) and II responding in 퐺 ∗ (ℎ , ℎ ) 푛 푋푛 푉 푋푛 푋푛+1 푉 푋푛 푋푛+1 by copying I’s moves in 퐺 ∗ (ℎ , ℎ ). This uniquely determines for each 푉 푋푛+1 푋푛+2

43 2 Sequences in sets and orders

푖 푖 푛 a finite play (푌 , 푌 ) of the game 퐺 ∗ (ℎ , ℎ ) ending with some 푛 푛+1 푖⩽푙푛 푉 푋푛 푋푛+1 푙푛 푙푛 푖 푖 푌푛 ⩽̸ 푌 in 푄. Clearly the play (푌 , 푌 ) depends only on the value 푛+1 푛 푛+1 푖⩽푙푛 taken by ℎ on the 푋푗 with 푗 ∈ {푛, … , 푛+푙푛 +2}. By the rules of the game 퐺푉 ∗ 푙푛 푙푛+1 푋 푙0 푋 푙푛 푙푛+1 for every 푛 we have 푌푛+1 = 푌푛+1 . We let 푌0 = 푌0 and 푌푛+1 = 푌푛+1 = 푌푛+1 . ∞ 푋 푋 We define 푔 ∶ [휔] → 푄 by letting 푔(푋) = 푌0 . Since 푌0 depends only on ℎ푋 , … ℎ푋 and ℎ is locally constant, it follows that 푔 is locally constant. 0 푙0+2 푋 ∗ 푋 ⩽̸ Moreover, by construction 푔(∗푋) = 푌0 = 푌1 and so 푔(푋) 푔(∗푋). Corollary 2.56. If 푄 is bqo, then 푉 ∗(푄) is bqo.

We now briefly show that there is a strong converse to Corollary 2.56. Let 푓 ∶ 퐹 → 푄 be a super-sequence from a front on 휔 into a qo 푄. We define by recursion on the well-founded relation ⊐ on 퐹 a map 푓̃ ∶ 퐹 → 푉 ∗(푄) by

푓(푠)̃ = 푓(푠) if 푠 ∈ 퐹 , 푓(푠)̃ = {푓(푠̃ ∪ {푛}) ∣ 푛 ∈ 휔/푠 and 푠 ∪ {푛} ∈ 퐹 } otherwise.

As long as 퐹 is not trivial we have [휔]1 ⊆ 퐹 and restricting 푓 ̃to [휔]1 we obtain ̃ 1 ∗ ̃ the sequence 푓↾[휔]1 ∶ [휔] → 푉 (푄). Notice also that 푓(푠) ∈ 푄 if and only if 푠 ∈ 퐹 .

̃ ∗ Lemma 2.57. If 푓 ∶ 퐹 → 푄 is bad, then 푓↾[휔]1 is a bad sequence in 푉 (푄).

Proof. By way of contradiction suppose that for some 푚0, 푛0 ∈ 휔 with 푚0 < ̃ ̃ ∗ 푛0 we have 푓(푚0) ⩽ 푓(푛0) in 푉 (푄) and let 휎 be a winning strategy for ̃ ̃ Player II in 퐺푉 ∗ (푓(푚0), 푓(푛0)). Let 푠0 = (푚0), 푡0 = (푛0) and 푢0 = (푚0, 푛0). ̃ ̃ We consider the following play of 퐺푉 ∗ (푓(푚0), 푓(푛0)). Observe that if 푠0 = ̃ (푚0) ∉ 퐹 , then 푢0 = (푚0, 푛0) ∈ 퐹 . We make Player I start with 푓(푠1) where 푠1 = 푠0 if 푠0 ∈ 퐹 and 푠1 = 푢0 otherwise. Then II answers according to 휎 by ̃ 푓(푡1) for some 푡1 ∈ 퐹 . If 푡0 = (푛0) ∈ 퐹 , then necessarily 푡1 = 푡0 and we let ⌢ 푢1 = 푢0 (푘) with 푘 = 1 + max 푢0. Otherwise 푡0 ⊏ 푡1 and 푡1 = (푛0, 푛1) for ⌢ some 푛1 > 푛0, we then let 푢1 = 푢0 ∪ 푡1 = 푢0 (푛1). Notice that in any case ¡ 푠1 푡1 since for 푋 = 푢1 ∪ 휔/푢1 we have 푠1 ⊏ 푋 and 푡1 ⊏ ∗푋. Then we ̃ make I respond with 푓(푠2) where 푠2 = 푠1 if 푠1 ∈ 퐹 , 푠2 = 푢2 if 푠1 ∉ 퐹 . We continue in this fashion, an example of which is depicted in Figure 2.5. After finitely many rounds I has reached some 푓(푠) for 푠 ∈ 퐹 , and II has reached some 푓(푡) with 푡 ∈ 퐹 . By construction 푠 ¡ 푡, but since 휎 is winning for II, we have 푓(푠) ⩽ 푓(푡), a contradiction.

44 2 Sequences in sets and orders

∗ ̃ ̃ ⎩ { { { { { { 퐺{ 푉{ ({ 푓(푚{ ⎨ 0),{ 푓(푛{ { { 0)){ { { { { { ⎧

I II ⎧ 푚0 푛0 ⎫ { } 휎 { } 푓(푚̃ , 푛 ) 푓(푛̃ , 푛 ) { 푛 푛 } 0 0 0 1 { 0 1 } 휎 푠 = } ̃ ̃ ⎨ 푓(푚0, 푛0, 푛1) 푓(푛0, 푛1, 푛2) { 푛1 푛2 = 푡 { ⎬ 휎 { } 푓(푚̃ , 푛 , 푛 , 푛 ) 푓(푛̃ , 푛 , 푛 , 푛 ) { 푛 푛 } 0 0 1 2 0 1 2 3 ⎩ 2 3 } 휎 } ̃ ⩽ ̃ } 푓(푚0, 푛0, 푛1, 푛2) 푓(푛0, 푛1, 푛2, 푛3, 푛4) 푛4 ⎭ 푓(푠) ⩽ 푓(푡) 푠 ¡ 푡

Figure 2.5: Copying and shift.

Notice that by definition 푓̃ ∶ 퐹 → 푉 ∗(푄) only reaches hereditarily countable non-empty sets over 푄, namely the elements of 푄 and the countable non-empty sets of hereditarily countable non-empty sets over 푄. Let 퐻∗ (푄) denote the 휔1 set of hereditarily countable non-empty sets over 푄 equipped with the qo induced from 푉 ∗(푄). By the axiom of countable choice we have 푋 ∈ 퐻∗ (푄) 휔1 ∗ if and only if 푋 ∈ 푉 (푄) and the set tcQ(푋) is countable. We have obtained the well known equivalence.

Theorem 2.58. A quasi-order 푄 is bqo if and only if 퐻∗ (푄) is wqo. 휔1 Notice that by definition any countable non-empty subset of 퐻∗ (푄) belongs 휔1 to 퐻∗ (푄). Moreover, by Proposition 2.6 (W6) a quasi-order is wqo if and 휔1 only if the qo 풫 (푄) of its countable subsets is well-founded, so 퐻∗ (푄) is <ℵ1 휔1 wqo if and only if it is well-founded.

Theorem 2.59. A quasi-order 푄 is bqo if and only if 퐻∗ (푄) is well-founded. 휔1

2.3 Around the definition of better-quasi-order

In the previous section, we were led to the definition of bqos by reflecting a bad sequence in 푉 ∗(푄) into some bad multi-sequence in 푄. In this section, we discuss the definition we obtained and try to understand what its essential features are. Along this line we show that the presence of the shift is somewhat

45 2 Sequences in sets and orders accidental. The material of this section has not yet been published by the author.

2.3.1 The perfect versus bad dichotomy For every 푋 ∈ [휔]∞, let us denote invariably by 햲 ∶ [푋]∞ → [푋]∞ the shift ∞ map defined by 햲(푁) = ∗푁 for every 푁 ∈ [푋] . For our discussion, we wish to treat both the pairs ([푋]∞, 햲), for 푋 ∈ [휔]∞, and the quasi-orders (푄, ⩽) as objects in the same category. With this in mind, let us call a topological digraph a pair (퐴, 푅) consisting of a topological space 퐴 together with a binary relation 푅 on 퐴. If (퐴, 푅) and (퐵, 푆) are topological digraphs, a continuous homomorphism from (퐴, 푅) to (퐵, 푆) is a continuous map 휑 ∶ 퐴 → 퐵 such that for every 푎, 푎′ ∈ 퐴, 푎 푅 푎′ implies 휑(푎) 푆 휑(푎′). As an important particular case, if 푓 ∶ 퐴 → 퐴 is any function we write (퐴, 푓) for the topological digraph whose binary relation is the graph of the function 푓. If 푓 ∶ 퐴 → 퐴 and 푔 ∶ 퐵 → 퐵 are functions, a map 휑 ∶ 퐴 → 퐵 is a continuous homomorphism from (퐴, 푓) to (퐵, 푔) exactly in case 휑 is continuous and 휑 ∘ 푓 = 푔 ∘ 휑. For a binary relation 푅 on 퐴 let us denote by 푅∁ the binary relation (퐴 × 퐴) ∖ 푅. Observe that for a discrete space 퐴, a multi-sequence ℎ ∶ [푋]∞ → 퐴 is continuous exactly when it is locally constant. Proposition 2.60. Let 푓 ∶ [휔]∞ → [휔]∞ be a continuous map such that 푓(푋) ⊆ 푋 for every 푋 ∈ [휔]∞ and 푅 be a binary relation on a discrete space 퐴. For every continuous 휑 ∶ [휔]∞ → 퐴 there exists 푍 ∈ [휔]∞ such that either 휑 ∶ ([푍]∞, 푓) → (퐴, 푅) is a continuous homomorphism, or 휑 ∶ ([푍]∞, 푓) → (퐴, 푅∁) is a continuous homomorphism. Proof. Let 휑 ∶ [휔]∞ → (퐴, 푅) be locally constant and define 푐 ∶ [휔]∞ → 2 by 푐(푋) = 1 if and only if 휑(푋) 푅 휑(푓(푋)). Clearly 푐 is locally constant so let 푐↓ ∶ 퐹 푐 → 2 be the associated super-sequence. By Nash-Williams’ Theorem 2.36 ↓ 푐 there exists an infinite subset 푍 of 휔 such that 푐 ↾퐹 푐|푍 ∶ 퐹 |푍 → 2 is constant. ∞ Therefore for the restriction 휓 = 휑↾[푍]∞ ∶ [푍] → 퐴 it follows that either 휓 ∶ ([푍]∞, 푓) → (퐴, 푅∁) is a continuous homomorphism, or 휓 ∶ ([푍]∞, 푓) → (퐴, 푅) is a continuous homomorphism. Remark 2.61. The previous proposition generalises as follows. Let 퐴 be any topological space, 푅 ⊆ 퐴 × 퐴 be a Borel binary relation and 푓 ∶ [휔]∞ → [휔]∞ a Borel map such that 푓(푋) ⊆ 푋 for every 푋 ∈ [휔]∞. For every Borel map 휑 ∶ [휔]∞ → 퐴 there exists 푍 ∈ [휔]∞ such that

46 2 Sequences in sets and orders either 휑 ∶ ([푍]∞, 푓) → (퐴, 푅) is a Borel homomorphism, or 휑 ∶ ([푍]∞, 푓) → (퐴, 푅∁) is a Borel homomorphism. Indeed, the set

−1 {푋 ∈ [휔]∞ ∣ 휑(푋) 푅 휑(푓(푋))} = (휑 × (휑 ∘ 푓)) (푅) is Borel in [휔]∞ and thus, by the Galvin-Prikry theorem [GP73], there exists a 푍 ∈ [휔]∞ as required. Definition 2.62. Let 푅 be a binary relation on a discrete space 퐴. (1) A multi-sequence ℎ ∶ [푋]∞ → 퐴 is perfect if ℎ ∶ ([푋]∞, 햲) → (퐴, 푅) is a ∞ homomorphism, i.e. if ℎ(푁) 푅 ℎ(∗푁) for every 푁 ∈ [푋] , (2) A super-sequence 푓 ∶ 퐹 → 퐴 is perfect if for every 푠, 푡 ∈ 퐹 , 푠 ¡ 푡 implies 푓(푠) 푅 푓(푡). In particular letting 푓 = 햲 in Proposition 2.60, we obtain the following well-known equivalence. Corollary 2.63. For a quasi-order 푄 the following are equivalent. (i) 푄 is bqo, (ii) every locally constant multi-sequence in 푄 admits a perfect sub-multi- sequence. (iii) every super-sequence in 푄 admits a perfect sub-super-sequence. Proposition 2.60 also suggests the following generalisation of the notion of bqo to arbitrary relations: Definition 2.64. A binary relation 푅 on a discrete space 퐴 is a better-relation on 퐴 if there is no continuous homomorphism 휑 ∶ ([휔]∞, 햲) → (퐴, 푅∁). This definition first appeared in a paper by Shelah [She82] and plays an important rôle in a work by Marcone [Mar94]. Of course a better-quasi-order is simply a better-relation which is reflexive and transitive. Remark 2.65. One could also consider non discrete analogues of the notion of better-quasi-orders and better-relations. Louveau and Saint Raymond [LS90] define a topological better-quasi-order as a pair (퐴, ⩽), where 퐴 is a topological space and ⩽ is a quasi-order on 퐴, such that there is no Borel homomorphism 휑 ∶ ([휔]∞, 햲) → (퐴, ⩽∁). We believe that topological analogs of bqo and better-relations deserve further investigations.

47 2 Sequences in sets and orders

2.3.2 Generalised shifts The topological digraph ([휔]∞, 햲) is central to the definition of bqo. Indeed a qo 푄 is bqo if and only if there is no continuous morphism ℎ ∶ ([휔]∞, 햲) → (푄, ⩽∁). In general, one can ask for the following: Problem 1. Characterise the topological digraphs which can be substituted for ([휔]∞, 햲) in the definition of bqo.

Let us write (퐴, 푅) ⩽ch (퐵, 푆) if there exists a continuous homomorphism from (퐴, 푅) to (퐵, 푆) and (퐴, 푅) ≡ch (퐵, 푆) if both (퐴, 푅) ⩽ch (퐵, 푆) and (퐵, 푆) ⩽ch (퐴, 푅) hold. Notice that a binary relation 푆 on a discrete space 퐵 is a better-relation if ∞ ̸ ∁ and only if ([휔] , 햲)⩽ch(퐵, 푆 ). Therefore any topological digraph (퐴, 푅) with ∞ (퐴, 푅) ≡ch ([휔] , 햲) can be used in the definition of better-relation in place of ([휔]∞, 햲). We do not know whether the converse holds, namely if (퐴, 푅) is a topological digraph which can be substituted to ([휔]∞, 햲) in the definition of ∞ bqo, does it follow that (퐴, 푅) ≡ch ([휔] , 햲)? We now show that at least the shift map 햲 can be replaced by certain ‘generalised shifts’. To this end, we first observe that the topological space [휔]∞ admits a natural structure of monoid. Following Solecki [Sol13] and Prömel and Voigt [PV86], we use the language of increasing injections rather than that of sets. We denote by ℰ the monoid of embeddings of (휔, <) into itself under composition, ℰ = {푓 ∶ 휔 → 휔 ∣ 푓 is injective and increasing}.

∞ For every 푋 ∈ [휔] , we let 푓푋 ∈ ℰ denote the unique increasing and injective enumeration of 푋. Conversely we associate to each 푓 ∈ ℰ the infinite subset Im 푓 of 휔 consisting in the range of 푓. Therefore the set of substructures of (휔, <) which are isomorphic to the whole structure (휔, <), namely [휔]∞, is in one-to-one correspondence with the monoid of embeddings of (휔, <) into itself. Moreover observe that for all 푋, 푌 ∈ [휔]∞ we have

푋 ⊆ 푌 ⟷ ∃푔 ∈ ℰ 푓푋 = 푓푌 ∘ 푔, so the inclusion relation on [휔]∞ is naturally expressed in terms of the monoid operation. Also, the set [푋]∞ corresponds naturally to the following right ideal:

푓푋 ∘ ℰ = {푓푋 ∘ 푔 ∣ 푔 ∈ ℰ}. As for [휔]∞, ℰ is equipped with the topology induced by the Baire space 휔휔 of all functions from 휔 to 휔. In particular, the composition ∘ ∶ ℰ × ℰ → ℰ, (푓, 푔) ↦ 푓 ∘ 푔 is continuous for this topology.

48 2 Sequences in sets and orders

Observe now that, in the terminology of increasing injections, the shift map 햲 ∶ ℰ → ℰ is simply the composition on the right with the successor function 헌 ∈ ℰ, 헌(푛) = 푛 + 1. Indeed for every 푋

푓 = 푓 ∘ 헌. ∗푋 푋

This suggests to consider arbitrary injective increasing function 푔, 푔 ≠ id휔, in place of the successor function. For any 푔 ∈ ℰ, we write ⃗푔∶ ℰ → ℰ, 푓 ↦ 푓 ∘푔 for the composition on the right by 푔. In particular, 헌⃗ = 햲 is the usual shift and in our new terminology we have ([휔]∞, 햲) = (ℰ, 헌)⃗ . The main result of this section is that these generalised shifts ⃗푔 are all equivalent as far as the theory of better-relations is concerned.

Theorem 2.66. For every increasing injective function 푔 ∈ ℰ, with 푔 ≠ id휔, ∞ we have (ℰ, ⃗푔) ≡ch ([휔] , 햲). Theorem 2.66 follows from Lemmas 2.72 and 2.73 below, but let us first state explicitly some of the direct consequences. Remark 2.67. Every topological digraph (퐴, 푅) has an associated topological graph (퐴, 푅s) whose symmetric and irreflexive relation 푅s is given by

푎 푅s 푏 ⟷ 푎 ≠ 푏 and (푎 푅 푏 or 푏 푅 푎).

The chromatic number and Borel chromatic number of topological graphs are studied by Kechris, Solecki, and Todorčević [KST99]. Notably the associated ∞ graph of ([휔] , 햲) has chromatic number 2 and Borel chromatic number ℵ0 (see also the paper by Di Prisco and Todorčević [DT06]). It directly follows from Theorem 2.66 that for every 푔 ∈ ℰ, with 푔 ≠ id휔, the associated graph of (ℰ, ⃗푔) also has chromatic number 2 and Borel chromatic number ℵ0. Definition 2.68. Let 푔 ∈ ℰ, 푅 a binary relation on a discrete space 퐴. We say (퐴, 푅) is a 푔-better-relation if one of the following equivalent conditions hold:

(1) for every continuous 휑 ∶ ℰ → 퐴 there exists 푓 ∈ ℰ such that the restriction 휑푓 ∶ (푓 ∘ ℰ, ⃗푔) → (퐴, 푅) is a continuous morphism, (2) there is no continuous morphism 휑 ∶ (ℰ, ⃗푔) → (퐴, 푅∁).

In case ⩽ is a quasi-order on a discrete space 푄, we say that 푄 is 푔-bqo instead of (푄, ⩽) is a 푔-better-relation.

49 2 Sequences in sets and orders

Of course this notion trivialises for 푔 = id휔, since an id휔-better-relation is simply a reflexive relation. Moreover better relation corresponds to 헌-better- relation.

Theorem 2.69. Let 푔 ∈ ℰ ∖ {id휔}, 푅 a binary relation on a discrete space 퐴. Then 푅 is a 푔-better-relation if and only if 푅 is a better-relation. In particular, a quasi-order (푄, ⩽) is 푔-bqo if and only if (푄, ⩽) is bqo. Corollary 2.70. A qo 푄 is bqo if and only if for every locally constant 휑 ∶ ℰ → 푄 and every 푔 ∈ ℰ there exists 푓 ∈ ℰ such that 휑(푓) ⩽ 휑(푓 ∘ 푔). As a corollary we have the following strengthening of Corollary 2.63 which is obtained by repeated applications of Proposition 2.60. Proposition 2.71. Let 푄 be bqo and 휑 ∶ ℰ → 푄 be locally constant. For every finite subset 풢 of ℰ there exists ℎ ∈ ℰ such that the restriction 휑 ∶ ℎ ∘ ℰ → 푄 is perfect with respect to every member of 풢, i.e. for every 푓 ∈ ℰ and every 푔 ∈ 풢 휑(ℎ ∘ 푓) ⩽ 휑(ℎ ∘ 푓 ∘ 푔). Getting a result of this kind was one of our motivations for proving The- orem 2.66. Indeed they are situations where we need to deal with an arbitrary super-sequence into some bqo and where going to a perfect sub-super-sequence is not enough (see Remark 4.44). We believe that Proposition 2.71 can still be improved. Finally here are the two lemmas which yield the proof of Theorem 2.66.

Lemma 2.72. Let 푔 ∈ ℰ ∖ {id휔}. Then (ℰ, ⃗푔) ⩽ch (ℰ, 헌)⃗ , i.e. there exists a continuous map 휌 ∶ ℰ → ℰ such that for every 푓 ∈ ℰ 휌(푓 ∘ 푔) = 휌(푓) ∘ 헌.

Proof. Since 푔 ≠ id휔, there exists 푘푔 = min{푘 ∈ 휔 ∣ 푘 < 푔(푘)}. Define 푛 0 푛+1 푛 퐺 ∶ 휔 → 휔 by 퐺(푛) = 푔 (푘푔), where 푔 = id휔 and 푔 = 푔 ∘ 푔 . Clearly 퐺 ∈ ℰ. We let 휌(푓) = 푓 ∘ 퐺 for every 푓 ∈ ℰ. The map 휌 ∶ ℰ → ℰ is continuous and for every 푓 ∈ ℰ and every 푛 we have

푛 휌(푓 ∘ 푔)(푛) = 푓 ∘ 푔 ∘ 푔 (푘푔) = 푓 ∘ 퐺(푛 + 1) = (휌(푓) ∘ 헌)(푛).

Lemma 2.73. Let 푔 ∈ ℰ ∖ {id휔}. Then (ℰ, 헌)⃗ ⩽ch (ℰ, ⃗푔), i.e. there exists a continuous map 휎 ∶ ℰ → ℰ such that for every 푓 ∈ ℰ 휎(푓 ∘ 헌) = 휎(푓) ∘ 푔.

50 2 Sequences in sets and orders

Proof. Let 푘푔 = min{푘 ∣ 푘 < 푔(푘)}. As in the proof of the previous Lemma 푛 we define 퐺 ∈ ℰ by 퐺(푛) = 푔 (푘푔). For every 푓 ∈ ℰ and every 푙 ∈ 휔, we let

푙 if 푙 < 퐺(0), 휎(푓)(푙) = { 푔푓(푛)−푛(푙) if 퐺(푛) ⩽ 푙 < 퐺(푛 + 1), for 푛 ∈ 휔.

Let us check that 휎(푓) is indeed an increasing injection from 휔 to 휔 for every 푓 ∈ ℰ. Since 휎(푓) is increasing and injective on each piece of its definition, it is enough to make the two following observations. Firstly, if 푙 < 퐺(0), then

휎(푓)(푙) = 푙 < 퐺(0) ⩽ 퐺 ∘ 푓(0) = 푔푓(0)(퐺(0)) = 휎(푓)(퐺(0)).

Secondly, if 퐺(푛) ⩽ 푙 < 퐺(푛 + 1) then

휎(푓)(푙) = 푔푓(푛)−푛(푙) < 푔푓(푛)−푛(퐺(푛 + 1)) 푓(푛)+1 푓(푛+1) = 푔 (푘푔) ⩽ 푔 (푘푔) = 퐺(푓(푛 + 1)), but we have

퐺(푓(푛 + 1)) = 푔푓(푛+1)−(푛+1)(퐺(푛 + 1)) = 휎(푓)(퐺(푛 + 1)).

One can easily check that 휎 ∶ ℰ → ℰ is continuous. Now on the one hand

푙 if 푙 < 퐺(0), 휎(푓 ∘ 헌)(푙) = { 푔푓(푛+1)−푛(푙) if 퐺(푛) ⩽ 푙 < 퐺(푛 + 1), for 푛 ∈ 휔. and on the other hand

푔(푙) if 푔(푙) < 퐺(0), 휎(푓)(푔(푙)) = { 푔푓(푛)−푛(푔(푙)) if 퐺(푛) ⩽ 푔(푙) < 퐺(푛 + 1).

By definition of 퐺, we have 푔(푙) < 퐺(0) if and only if 푙 = 푔(푙). Moreover if 퐺(푛) ⩽ 푙 < 퐺(푛 + 1) then we have 퐺(푛 + 1) ⩽ 푔(푙) < 퐺(푛 + 2) and so

휎(푓 ∘ 헌)(푙) = 푔푓(푛+1)−푛(푙) = 푔푓(푛+1)−(푛+1)(푔(푙)) = 휎(푓)(푔(푙)), which proves the Lemma.

51 3 Sequences in spaces

This chapter is dedicated to the sequences and super-sequences in metric spaces and topological spaces. The results presented here appear in an article [CP14] published by the author and R. Carroy in Fundamenta Mathematicae. We start by making some simple observations on Cauchy sequences in metric spaces. Recall that a sequence (푥푛)푛∈휔 into a metric space (풳, 푑풳) is Cauchy if for every 휀 > 0 there exists 푘 ∈ 휔 such that for every 푚, 푛 ⩾ 푘 we have 푑풳(푥푚, 푥푛) < 휀. As we have already done we consider sequences as maps from the front [휔]1. But we now turn [휔]1 into a metric space by considering this front as a subset of the Cantor space. The Cantor space is the product space 2휔 where 2 is the discrete two points space. A base of clopen sets of 2휔 is given by the sets of the form

휔 <휔 푁푢 = {푥 ∈ 2 ∣ 푢 ⊏ 푥} for 푢 ∈ 2 . For a point 푥 ∈ 2휔 a neighbourhood base is given by the sets 푁 for 푛 ∈ 휔. 푥↾푛 The space 2휔 admits the compatible metric given by 푑(푥, 푦) = 2−푛−1 if 푥 ≠ 푦 and 푛 is the least natural number with 푥(푛) ≠ 푦(푛). This metric is complete and 2휔 is a very important example of Polish space, i.e. a separable completely metrisable topological space. We refer the reader to the monograph by Kechris [Kec95] for basic results about the Cantor space and Polish spaces in general. The Cantor space is compact and zero-dimensional. We identify each {푛} with its characteristic function 0푛10휔 in 2휔, therefore turning [휔]1 into a subset of 2휔. This makes [휔]1 a metric space. Observe that for every positive natural number 푘,

푑({푚}, {푛}) < 2−푘−1 ⟷ either 푚 = 푛, or 푚 > 푘 and 푛 > 푘.

We recall that a function 푓 ∶ (풳, 푑풳) → (풴, 푑풴) between metric spaces is uniformly continuous if for every 휀 > 0 there exists 훿 > 0 such that for every ′ ′ ′ 푥, 푥 ∈ 풳 we have 푑풳(푥, 푥 ) < 훿 implies 푑풴(푓(푥), 푓(푥 )) < 휀.

Fact 3.1. A sequence (푥푛)푛∈휔 in a metric space (풳, 푑풳) is Cauchy if and only 1 if the mapping [휔] → 풳, {푛} ↦ 푥푛 is uniformly continuous.

52 3 Sequences in spaces

Proof. Suppose that for every 휀 > 0 there exists 훿 > 0 such that for every 푚, 푛 ∈ 휔, 푑({푛}, {푚}) < 훿 implies 푑풳(푥푛, 푥푚) < 휀. Then for a positive −푘−1 natural number 푘 with 2 < 훿 we have 푛, 푚 > 푘 implies 푑풳(푥푛, 푥푚) < 휀, so (푥푛) is Cauchy. Conversely suppose (푥푛) is Cauchy and let 휀 > 0. There exists 푘 ∈ 휔 such −푘−1 that 푚, 푛 > 푘 implies 푑풳(푥푚, 푥푛) < 휀. So if we take 훿 = 2 , then for every 푚, 푛 ∈ 휔, 푑({푚}, {푛}) < 훿 implies 푑풳(푥푚, 푥푛) < 휀. For any metric space 풳 there exists a unique – up to isometry – complete metric space 풳̂ such that 풳 is a dense subspace of 풳̂, called the metric completion of 풳. The metric completion of 풳 is characterised by following universal property: every uniformly continuous map 푓 ∶ 풳 → 풴 into a complete metric space extends to a unique continuous map 푓̂ ∶ 풳̂ → 풴. Recall that a metric space 풳 is totally bounded if for every 휀 > 0 there exists a covering of 풳 by finitely many open balls of radius 휀. A metric space is totally bounded if and only if its metric completion is compact. By the Heine-Cantor theorem ([Kec95, (4.5), p. 19]) any continuous map from a compact metric space into a metric space is uniformly continuous. It follows that a map 푓 ∶ 풳 → 풴 from a totally bounded metric space 풳 into a complete metric space 풴 is uniformly continuous if and only if it extends to unique continuous map 푓̂ ∶ 풳̂ → 풴. The metric completion of [휔]1 is simply given by its closure inside 2휔, namely the set {0푛10휔 ∣ 푛 ∈ 휔} ∪ {0휔} which we identify with [휔]⩽1. Of course, a sequence into a complete metric space 풳 is Cauchy if and only if it converges. 1 Or in different words, a sequence (푥푛)푛∈휔 is Cauchy if and only if [휔] → 풳, ⩽1 {푛} ↦ 푥푛, extends to a continuous map [휔] → 풳.

3.1 Cauchy super-sequences

As we did for the front [휔]1 we now view every front as a subsets of the Cantor space.

휔 Notation 3.2. For any subset 푢 of 휔 we let 휒푢 ∈ 2 be the characteristic 휔 휔 function of 푢 on 휔. So for instance, 휒{2,4} = 001010 and 휒∅ = 0 . For a finite subset 푠 of 휔, the restricted characteristic function of 푠, denoted by

̃휒푠, is 휒푠↾max 푠+1 if 푠 ≠ ∅ and the empty sequence otherwise. For instance, ̃휒{2,4} = 00101.

<∞ We embed every subset of [휔] into the Cantor space via 푠 ↦ 휒푠. By misuse of language, we sometimes identify a subset of 휔 with its characteristic function inside the Cantor space. In the same way, families of finite sets of

53 3 Sequences in spaces natural numbers are sometimes identified with the corresponding subset of the Cantor space. Every front 퐹 is therefore considered a metric space for the restriction of the metric on the Cantor space. With the introductory discussion in mind, we make the following definition.

Definition 3.3. Let 풳 be a metric space. A super-sequence 푓 ∶ 퐹 → 풳 is said to be Cauchy if it is uniformly continuous.

For a family 퐹 ⊆ [휔]<∞ the closure of 퐹 , denoted by 퐹 , is the topological closure of the set {휒푠 ∣ 푠 ∈ 퐹 } inside the Cantor space. Of course, the closure of a family 퐹 coincides with the metric completion of 퐹 . Therefore every Cauchy super-sequence 푓 ∶ 퐹 → 풳 into a complete metric space 풳 extends to a continuous map 푓̄ ∶ 퐹 → 풳. Examples 3.4. (a) The map 푓 ∶ [휔]2 → 2휔 defined by

푓(푚, 푛) = 0푚10푛10휔

is Cauchy. It extends to the continuous map 푓̄ ∶ [휔]⩽2 → 2휔 where 푓({푚})̄ = 0푚10휔 for every 푚 ∈ 휔 and 푓(∅) = 0휔.

(b) The map 푓 ∶ [휔]2 → 2휔 defined by

0휔 if 푛 is even, 푓(푚, 푛) = { 1휔 otherwise,

is not Cauchy.

1휔 if 푚 + 1 = 푛, 푓(푚, 푛) = { 0푚10휔 otherwise,

is not Cauchy. Notice however that for every 푚 the sequence 푓(푚, 푛) converges trivially to 0푚10휔 when 푛 tends towards infinity. Moreover the sequence 0푚10휔 converges to 0휔 when 푚 tends towards infinity. How- ever the sequence 푓(푚, 푚 + 1) is constant equal to 1휔 while (푚, 푚 + 1) converges to ∅ in 2휔. Therefore there is no continuous extension of 푓 to the closure of [휔]2 and so 푓 is not Cauchy. For a front 퐹 , the closure 퐹 is closely related to the tree 푇 (퐹 ) associated to 퐹 (cf. Lemma 2.29). The following result is stated by Todorčević [Tod10].

54 3 Sequences in spaces

Proposition 3.5. For every front 퐹 we have

휔 퐹 = {휒푠 ∈ 2 ∣ 푠 ∈ 푇 (퐹 )}. Proof. Let 퐹 be a front on 푋. ⊇: Suppose 푠 is in 푇 (퐹 ) ∖ 퐹 , i.e. 푠 ⊏ 푡 for some 푡 ∈ 퐹 . For every 푛 > max 푠 there is a 푢 ∈ 퐹 with 푢 ⊏ 푠∪푋/푛. If 푢 ⊑ 푠 then we have 퐹 ∋ 푢 ⊏ 푡 ∈ 퐹 , a contradiction. Hence 푠 ⊏ 푢 and we found a 푢 ∈ 퐹 with 휒 ∈ 푁 . 푢 휒푠↾푛 Since 푛 was arbitrarily large, it follows that 휒푠 ∈ 퐹 . ⊆: Conversely suppose that an element 푥 of 2휔 belongs to 퐹 . We first show that 푥 is the characteristic function of a finite subset of 푋. Since 푋 휔 푋 2 = {휒퐸 ∣ 퐸 ∈ 풫(푋)} is closed in 2 and 퐹 ⊆ 2 , necessarily 푥 is the characteristic function of a subset of 푋. Now suppose towards a contradiction that 푥 is the characteristic function of an infinite set 푀 ⊆ 푋. Then for all finite prefix 푢 of 푀 there exists 푠 ∈ 퐹 such that 휒 ∈ 푁 , and hence 푢 ⊑ 푠. But then {푢 ∣ 푢 ⊏ 푀} would be an 푠 푥↾max(푢)+1 infinite branch of 푇 (퐵), contradicting Lemma 2.29. <∞ Hence 푥 = 휒푠 for some 푠 ∈ [푋] . It only remains to show that there exists a 푡 ∈ 퐹 with 푠 ⊑ 푡. Pick 푛 > max 푠. Since 휒푠 belongs to the closure of 퐹 there is a 푡 ∈ 퐹 with 휒푠↾푛 ⊏ 휒푡, and so 푠 ⊑ 푡. Remark 3.6. The closure 퐹 of a front 퐹 , being a countable compact metrisable space, admits a natural rank, namely the Cantor-Bendixson rank, denoted by ∗ ∞ |퐹 |CB (see [Kec95, 6.C, p.33]). For every 푋 ∈ [휔] the closure of the front [푋]푘 has Cantor-Bendixson rank 푘. Moreover the Schreier barrier 풮 has Cantor- Bendixson rank 휔. From these examples one could think that for every front 퐹 the Cantor- Bendixson rank of 퐹 is equal to the rank of 퐹 . Contrary to what is said by Todorčević [Tod10, Definition 1.24, p.3], this is not the case in general asthe following example shows. Let 퐹 (0) = {푠 ∈ [휔/0]<∞ ∣ min 푠 = |푠|}, and 퐹 (푛) = [휔/푛]푛 for 푛 ⩾ 1.

Then we build the front 퐹 = seq푛∈휔 퐹 (푛) (cf. Figure 3.1). In fact 퐹 is even a barrier, namely whenever 푠 ⊆ 푡 in 퐹 then 푠 = 푡. Now we have rk(퐹 ) = 휔 + 1. However the 휔th Cantor-Bendixson derivative of 퐹 is equal to {∅, {0}}, and so ∗ we have |퐹 |CB = 휔. And notably the Cantor-Bendixson rank of 퐹0 is also 휔. It also follows from this example that when proving a result about fronts by induction on the Cantor-Bendixson rank of the closure we cannot in general apply the induction hypothesis to the rays. This is the main reason why we prefer rk 퐹 to the Cantor-Bendixson rank of 퐹 .

55 3 Sequences in spaces

⋯

∗ Figure 3.1: A front 퐹 with rk 퐹 = 휔 + 1 and |퐹 |CB = 휔.

We also note that the closure operation behaves nicely with respect to the taking of restrictions and rays.

Corollary 3.7. Let 퐹 be a front on 푋.

1. For all 푀 ∈ [푋]∞ we have 퐹 |푀 = 퐹 |푀;

2. For all 푛 ∈ 푋 we have 퐹 = (퐹 ) . 푛 푛 Proof. (1) It is enough to prove that 퐹 |푀 ⊆ 퐹 |푀 holds. So let 푠 ⊏ 푡 ∈ 퐹 with 푠 a subset of 푀. Since 퐹 is a front there is a 푢 ∈ 퐹 with 푢 ⊏ 푠∪푀/푠 and necessarily 푢 ∈ 퐹 |푀. If 푢 ⊑ 푠 ⊏ 푡 we have a contradiction. Hence 푠 ⊑ 푢 ∈ 퐹 |푀 and so 푠 ∈ 퐹 |푀 by Proposition 3.5.

(2) Since 퐹푛 is a front on 휔/푠, 퐹푛 = {휒푡 ∣ ∃푢 ∈ 퐹푛 푡 ⊑ 푢} by Proposition 3.5. Now if 푡 ⊑ 푢 ∈ 퐹 then {푛} ∪ 푡 ⊑ {푛} ∪ 푢 ∈ 퐹 and thus 푡 ∈ (퐹 ) . 푛 푛 Conversely if 푡 ∈ (퐹 ) then {푛} ∪ 푡 ∈ 퐹 and thus there exists 푢 ∈ 퐹 푛 with {푛} ∪ 푡 ⊑ 푢 ∈ 퐹 . Now 푡 ⊑ ∗푢 ∈ 퐹푛 and therefore 푡 ∈ 퐹푛. Of course a metric space is compact if and only if every sequence admits a converging subsequence. Similarly, a metric space is totally bounded if and only if every sequence admits a Cauchy subsequence. The main result of this chapter (cf. Section 3.3) is the following:

Theorem 3.8. Let 풳 be a compact metric space. Then every super-sequence in 풳 admits a Cauchy sub-super-sequence.

Using the fact that every non empty compact metric space 풳 is a continuous image of 2휔, it will be enough to prove the theorem in the zero-dimensional case. Namely we will prove the following (cf. Theorem 3.25).

56 3 Sequences in spaces

Theorem 3.9 (with R. Carroy). Every super-sequence into 2휔 admits a Cauchy sub-super-sequence. The reason for focusing in the zero-dimensional case stems from the existence of a nice and simple characterisation of uniform continuity. Let us observe this already in the simple case of a sequence. First we recall the following easy characterisation of the clopen sets of 2휔. Fact 3.10. A subset 퐶 of 2휔 is clopen if and only if there exists a finite set 퐴 ⊆ 2<휔 such that

(i) 푠, 푡 ∈ 퐴 implies 푠⊑푡̸ , (ii) 퐶 = ⋃ 푁 . 푠∈퐴 푠 Proof. Since each 푁 is clopen, if 퐶 = ⋃ 푁 for a finite set 퐴 then 퐶 푠 푠∈퐴 푠 is clopen. Conversely, if 퐶 is clopen in 2휔 then 퐶 = ⋃ 푁 for some set 푠∈퐵 푠 휔 퐵 ⊆ 2 since the 푁푠 form a base and 퐶 is open. Now as 퐶 is also closed in the compact space 2휔 it is compact and so there exists a finite set 퐵′ ⊆ 퐵 such that 퐶 = ⋃ 푁 . We can therefore take 퐴 to be the ⊑-minimal elements of 푠∈퐵′ 푠 퐵′. Now we can express uniform continuity of a sequence into a compact metric space in a simple way, reminiscent of topological continuity.

휔 Fact 3.11. Let (푥푛)푛∈휔 be a sequence in 2 . The following are equivalent:

(i) (푥푛)푛∈휔 is Cauchy; 1 휔 (ii) the map 푓 ∶ [휔] → 2 , 푛 ↦ 푥푛 is uniformly continuous; −1 휔 (iii) 푓 (퐶) = {푛 ∈ 휔 ∣ 푥푛 ∈ 퐶} is finite or cofinite for all clopen 퐶 of 2 . 휔 Proof. Suppose (푥푛)푛∈휔 is Cauchy and fix a clopen 퐶 of 2 . We can write 퐶 = ⋃ 푁 for a finite ⊑-antichain 퐴 ⊆ 2<휔, and we let 푘 = max{|푠| ∣ 푠 ∈ 푠∈퐴 푠 −푘−1 퐴}. There exists 푙 ∈ 휔 such that for all 푚, 푛 ⩾ 푙 we have 푑(푥푚, 푥푛) < 2 . By definition of the distance this means that 푥푚↾푘 = 푥푛↾푘 for all 푚, 푛 ⩾ 푙. Depending on whether 푥푙↾푘 ∈ 퐴 or not, we get that {푛 ∈ 휔 ∣ 푥푛 ∈ 퐶} is either cofinite or finite. 1 휔 Conversely, suppose that 푓 ∶ [휔] → 2 is such that {푛 ∈ 휔 ∣ 푥푛 ∈ 퐶} is finite or cofinite for all clopen 퐶 of 2휔. Let 푘 ∈ 휔 and consider the partition [휔]1 = ⋃ 푓−1(푁 ) into preimage of clopen sets. There exists a unique 푠 ∈ 푠∈2푘 푠 푘 −1 2 such that 푓 (푁푠) is cofinite. Hence there exists 푙 ∈ 휔 such that for all 푛 ⩾ 푙 −푘−1 we have 푠 ⊑ 푥푛 and therefore for all 푚, 푛 ⩾ 푙 we have 푑(푥푚, 푥푛) < 2 .

57 3 Sequences in spaces

Even though we defined at first Cauchy super-sequences in the setting of metric spaces, this notion is really pertaining to the ‘uniform structure’ of a front. Our next task is therefore to study the uniform structure of a subset of some compact zero-dimensional space.

3.2 Uniform subspaces of Boolean spaces

Les espaces métriques sont des « espaces uniformes » de nature par- ticulière; les espaces uniformes n’ont été définis de manière générale qu’en 1937, par A. Weil (XI). Auparavant on ne savait utiliser les notions et les résultats relatifs à « structure uniforme » que lor- squ’il s’agissait d’espaces métriques: ce qui explique le rôle important joué dans beaucoup de travaux sur la topologie, par les espaces métriques ou métrisables (et en particulier par les espaces compact métrisables) dans des questions où la distance n’est d’aucune utilité véritable. Nicolas Bourbaki [Bou06, TG II.43] The basic reference on uniform spaces is due to Bourbaki [Bou06] and we follow their terminology. A recent account on this topic is provided by Page [Pag11]. Every metric space (풳, 푑) has an associated uniform structure generated by the entourages of the form {(푥, 푦) ∈ 풳 × 풳 ∣ 푑(푥, 푦) < 휀} for some 휀 > 0. Very importantly, any compact Hausdorff topological space admits a unique uniform structure that agrees with its topology [Bou06, Theorem 1, II.27]. In particular, every Boolean space, i.e. compact Hausdorff zero-dimensional space, is unambiguously seen as a uniform space. In this section, we focus on those uniform spaces which arise as a uniform subspace of some Boolean space. We prove a simple characterisation of uniform continuity between such uniform spaces, which is reminiscent of topological continuity. This characterisation is of crucial importance in the next section. The following notion simplifies greatly the study of these uniform subspaces. Definition 3.12. Let 푆 be a subset of a Boolean space 풳. A subset 퐵 of 푆 is called a block of 푆 (relatively to 풳) if there exists a clopen 퐶 of 풳 such that 퐵 = 퐶 ∩ 푆. We write Blocks(푆) for the Boolean subalgebra of 풫(푆) of blocks of 푆. The uniform structure of a uniform subspace of a Boolean space 풳 is essentially given by its blocks as we show in Lemma 3.17 below (see also [Bou06,

58 3 Sequences in spaces

Exercice 12, II.38]). As a consequence, uniform continuity between such spaces admits of the following simple characterisation.

Proposition 3.13. Let 풳 and 풴 be two Boolean spaces, and let 푆 ⊆ 풳 and 푇 ⊆ 풴 be endowed with the induced uniform structure. Then a function 푓 ∶ 푆 → 푇 is uniformly continuous if and only if for all 퐵 ∈ Blocks(푇 ) we have 푓−1(퐵) ∈ Blocks(푆).

When a subset 푆 of a Boolean space 풳 is endowed with the induced uniform structure, then the completion of the uniform space 푆 coincide with the closure 푆 of 푆 in 풳1. Therefore a function 푓 ∶ 푆 → 푇 as in Proposition 3.13 is uniformly continuous if and only if there exists a unique continuous map 푓 ∶

푆 → 푇 such that 푓↾푆 = 푓. Although Proposition 3.13 is folklore, we did not find any reference for this very statement (see however the work of Gehrke, Grigorieff, and Pin [GGP10] in relation with automata theory and the one by Erné [Ern01]). To keep our exposition self-contained we now provide the reader with a series of lemmas that lead to a proof of this fact.

Lemma 3.14. Let 풳 be a Boolean space. The unique compatible uniform structure on 풳 admits

(i) as a base the entourages of the form 푈 = ⋃ 퐶 × 퐶 where (퐶 ) is a (퐶푖) 푖 푖 푖 푖 finite partition of 풳 into clopen sets;

(ii) as a subbase the entourages of the form 푈퐶 = (퐶 × 퐶) ∪ (풳 ∖ 퐶 × 풳 ∖ 퐶) where 퐶 is a clopen set of 풳.

Proof. Since 풳 is compact and Hausdorff, the unique uniform structure which is compatible with the topology of 풳 consists of the neighbourhoods of the diagonal of 풳 ([Bou06, Theorem 1, II.27]). Clearly each 푈 with (퐶 ) a (퐶푖) 푖 clopen partition of 풳 is a clopen neighbourhood of the diagonal. Conversely let 푈 be a neighbourhood of the diagonal of 풳. Since 풳 is zero-dimensional, there exists for each 푥 ∈ 풳 a clopen 퐶푥 such that 푥 ∈ 퐶푥 and 퐶푥 ×퐶푥 ⊆ 푈. By compactness of 풳, there exist 푥 , … , 푥 such that 퐶 , … , 퐶 covers 풳. We 1 푛 푥1 푥푛 can then construct a partition 퐶 , … , 퐶 of 풳 in clopen such that 퐶 ⊆ 퐶 . 1 푛 푖 푥푖 It follows that 푈 ⊆ 푈 which concludes the proof. (퐶푖) Lemma 3.15. Let 풳 be a Boolean space and let 퐹 be a closed subspace of 풳. Then the clopen sets of 퐹 coincide with the blocks of 퐹 .

1The space 푆 is also the Stone dual of the Boolean algebra Blocks(푆), cf. Subsection 4.1.3.

59 3 Sequences in spaces

Proof. By definition, if 퐵 is a block of 퐹 then 퐵 is clopen in 퐹 . Conversely, let 퐶 be clopen in 퐹 . The blocks of 퐹 form a clopen base of 퐹 , and since 퐶 is open, it is a union of a family ℬ of blocks of 퐹 . Since 퐹 is compact and 퐶 closed in 퐹 , there is a finite subset ℬ′ of ℬ for which 퐶 = ⋃ ℬ′. Since the blocks of 퐹 form a Boolean algebra, it follows that 퐶 is a block.

Lemma 3.16. Let 풳 be a Boolean space, and let 푆 ⊆ 푇 ⊆ 풳. Then

Blocks(푆) = {퐵 ∩ 푆 ∣ 퐵 ∈ Blocks(푇 )}.

Proof. Let 퐵 ∈ Blocks(푇 ) and let 퐶 be clopen in 풳 such that 퐵 = 퐶 ∩ 푇 . Then 퐵 ∩ 푆 = 퐶 ∩ 푆 is a block of 푆. Conversely, if 퐵 is a block of 푆 then there is 퐶 clopen in 풳 with 퐵 = 퐶 ∩ 푆. It follows that 퐵 = (퐶 ∩ 푇 ) ∩ 푆 where 퐶 ∩ 푇 ∈ Blocks(푇 ).

Lemma 3.17. Let 풳 be a Boolean space and let 푆 be a subset of 풳. Then the uniform structure induced on 푆 by 풳 admits

(i) as a base the entourages of the form 푈 = ⋃ 퐶 × 퐶 where (퐶 ) is a (퐶푖) 푖 푖 푖 푖 finite partition of 푆 into blocks;

(ii) as a subbase the entourages of the form 푈퐶 = (퐶 × 퐶) ∪ (푆 ∖ 퐶 × 푆 ∖ 퐶) where 퐶 is a block of 푆.

Proof. Applying Lemmas 3.14 and 3.15 we obtain that the uniform subspace 푆 admits as a subbase the entourages of the form 푈퐶 where 퐶 is a block of 푆. Moreover, the uniform subspace 푆 of 풳 is identical to the uniform subspace 푆 of 푆. It follows that entourages of the form (푆 × 푆) ∩ 푈퐶 where 퐶 is a block of 푆 constitute a subbase for the uniform space 푆. On the one hand (푆 × 푆) ∩ 푈퐶 = {(푠, 푡) ∈ 푆 × 푆 ∣ 푠 ∈ 퐶 ↔ 푡 ∈ 퐶} = 푈퐶∩푆. On the other hand, {퐶 ∩ 푆 ∣ 퐶 ∈ Blocks(푆)} = Blocks(푆) by Lemma 3.16. We have thus obtained that the entourages of the form 푈퐵 where 퐵 ∈ Blocks(푆) constitute a subbase for the uniform space 푆. Proof of Proposition 3.13. ⇒: Suppose 푓 is uniformly continuous and let 푓̂ ∶ 푆 → 푇 be its continuous extension. Then for all clopen 퐶 of 풴 the set 푓−1(퐶 ∩ 푇 ) = 푓−1̂ (퐶) ∩ 푆 is a block of 푆. ⇐: Suppose 푓 ∶ 푆 → 푇 preserves blocks by preimage. By Lemma 3.17, it is enough to show that for each block 퐵 of 푇 the preimage of 푈퐵 by 푓 × 푓 is an −1 entourage of 푆. In fact, (푓 × 푓) (푈퐵) = 푈푓−1(퐵).

60 3 Sequences in spaces

3.3 Cauchy super-sequences

We endow every front 퐹 with the uniform structure inherited from the Cantor space and Blocks(퐹 ) denotes the Boolean algebra of blocks of 퐹 :

Blocks(퐹 ) = {퐶 ∩ 퐹 ∣ 퐶 is a clopen of 2휔}.

For an arbitrary family 퐻 ⊆ [휔]<∞ and 푛 ∈ 휔, we recall that the ray of 퐻 at 푛 is the family

<∞ 퐻푛 = {푢 ∈ [휔/푛] ∣ {푛} ∪ 푢 ∈ 퐻}

Examples 3.18. (a) For the front [휔]1, the Boolean algebra Blocks([휔]1) consists of the finite or cofinite subsets of [휔]1 (cf. Fact 3.11).

(b) The Boolean algebra of blocks of the front [휔]2 consists of the subsets 퐵 2 of [휔] such that for every 푛 ∈ 휔 the ray 퐵푛 is either finite or cofinite and 1 either퐵푛 is empty for cofinitely many 푛, or 퐵푛 = [휔/푛] for cofinitely any 푛.

Figure 3.2: A block of the front [휔]2.

The ray of a family 퐻 ⊆ [휔]<∞ at some 푛 ∈ 휔 is closely related to the following subfamily of 퐻

퐻 = {푢 ∈ 퐻 ∣ {푛} ⊑ 푢} = {휒 ∈ 퐻 ∣ ̃휒 ⊏ 휒 } = 퐻 ∩ 푁 . 푛⊑ 푢 {푛} 푢 ̃휒{푛}

Lemma 3.19. Let 퐹 be a front on 푁, 푛 ∈ 푁 and 퐵 ⊆ 퐹 . Then 퐵푛⊑ ∈ Blocks(퐹 ) if and only if 퐵푛 ∈ Blocks(퐹푛).

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Proof. Notice first that 퐵푛⊑ = 퐵 ∩ 푁0푛1 and that 퐹푛 ⊆ 푁0푛+1 . Let 휂푛 ∶ 푁0푛+1 → 푁0푛1 be the homeomorphism given by 휂푛(푆) = {푛} ∪ 푆 for every

푆 ∈ 풫(휔/푛) = 푁0푛+1 . Suppose that 퐵푛⊑ ∈ Blocks(퐹 ) and let 퐶 ⊆ 푁0푛1 be 휔 −1 −1 clopen in 2 with 퐵푛⊑ = 퐹 ∩퐶. Then 휂푛 (퐶) is clopen and 퐵푛 = 휂푛 (퐶)∩퐹푛, so 퐵푛 is a block of 퐹푛. Conversely suppose that 퐵푛 ∈ Blocks(퐹푛) and let 휔 퐶 ⊆ 푁0푛+1 be clopen in 2 with 퐵푛 = 퐹푛 ∩ 퐶. Then 퐵푛⊑ = 퐹 ∩ 휂푛(퐶), and as 휂푛(퐶) is clopen we have 퐵푛⊑ ∈ Blocks(퐹 ). The following proposition gives a useful characterisation of the family of blocks of a front. Proposition 3.20. We have the following. (i) If 퐹 is the trivial front, then Blocks(퐹 ) = 풫(퐹 ).

(ii) If 퐹 is a non-trivial front and 퐵 is a subset of 퐹 , then 퐵 ∈ Blocks(퐹 ) if and only if

(a) 퐵푛 is a block of 퐹푛 for every 푛 ∈ 푁, and

(b) either for all but finitely many 푛 ∈ 푁 we have 퐵푛 = ∅, or for all but finitely many 푛 ∈ 푁 we have 퐵푛 = 퐹푛. Proof. The statement (i) is trivial. For (ii), suppose 퐹 is a non-trivial front and assume that 퐵 ⊆ 퐹 satisfies Conditions (a) and (b). For every 푛 ∈ 푁 the set 퐵푚 is a block of 퐹푚 and so by Lemma 3.19 퐵푚⊑ is a block of 퐹 . Notice that 퐵 is equal to the disjoint union of the 퐵푛⊑ for 푛 ∈ 푁. If 퐵푛 is empty for all but finitely many 푛, then 퐵 is a block as a finite union of blocks. Now if there exists 푘 such that 퐵푛 = 퐹푛 for every 푛 ∈ 푁/푘, then

퐵 = ⋃ 퐵푚⊑ ∪ 퐹 |(푁/푘) 푚∈푁 푚⩽푘 is also a block as a finite union of blocks, since 퐹 |(푁/푘) = 퐹 ∩ 푁0푘+1 . Conversely, if 퐵 is a block of 퐹 then there exists a clopen 퐶 of 2휔 with

퐵 = 퐹 ∩ 퐶. In particular 퐵푛⊑ = 퐹 ∩ 퐶 ∩ 푁0푛1 is a block of 퐹 for every 푛 ∈ 푁, and so 퐵푛 is a block of 퐹푛 by Lemma 3.19. Now we can write the clopen 퐶 as ⋃ 푁 for a finite ⊑-antichain 퐴 ⊆ 2<휔. Let 퐴 = {푠 ∈ 퐴 ∣ 0푛1 ⊑ 푠} for 푠∈퐴 푠 푛 every 푛 ∈ 휔 and notice that 퐵푛⊑ = 퐹 ∩ ⋃ 푁푠. Then either 퐴 = ⋃ 퐴푛 푠∈퐴푛 푛∈휔 or there exists 푙 such that 퐴 = {0푙}∪⋃ 퐴 . In the first case, it follows that 푛∈휔 푛 퐵푛 is empty for all but finitely many 푛 ∈ 푁. In the second case, it follows that 퐵푛 = 퐹푛 for every 푛 > 푙. In both cases the set 퐵 fulfils Conditions (a) and (b).

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Remark 3.21. The previous Proposition actually provides us with a definition of Blocks(퐹 ) for every front 퐹 by induction on the rank. Next we notice that the notion of block behaves nicely with the taking of restrictions. Observe that for 푆 ⊆ 푇 ⊆ [휔]<∞ and 푁 ∈ [휔]∞ we have

푆|푁 = 푆 ∩ [푁]<∞ = 푆 ∩ 2푁 = 푆 ∩ 푇 ∩ 2푁 = 푆 ∩ 푇 |푁, hence the restriction of 푆 to 푁 equals the trace of 푆 on 푇 |푁. Notation 3.22. Let 퐹 ⊆ [휔]<∞. For a family 퐒 of subsets of 퐹 and 푁 ∈ [휔]∞ we denote by 퐒|푁 the family

{푆|푁 ∣ 푆 ∈ 퐒} = {푆 ∩ 퐹 |푁 ∣ 푆 ∈ 퐒}.

In fact, the blocks of the restriction of a front are simply the restrictions of the blocks of the front. Lemma 3.23. Let 퐹 be a front on 푀 and let 푁 ⊆ 푀 be infinite. Then

Blocks(퐹 )|푁 = Blocks(퐹 |푁).

Proof. If 퐵 ∈ Blocks(퐹 ) then there exists 퐶 clopen in 2휔 with 퐵 = 퐶 ∩ 퐹 . It follows that

퐵|푁 = 퐵 ∩ 퐹 |푁 = 퐵 ∩ 퐹 ∩ 퐹 |푁 = 퐶 ∩ 퐹 |푁 is a block of 퐹 |푁. Conversely, if we have 퐵 = 퐶 ∩ 퐹 |푁 for some clopen 퐶 of 2휔 then for the block 퐵′ = 퐶 ∩ 퐹 of 퐹 we have 퐵′|푁 = 퐵′ ∩ 퐹 |푁 = 퐶 ∩ 퐹 ∩ 퐹 |푁 = 퐵. We observe that for a finite family 퐅 of subsets of a front 퐹 on 푀 we can find by repeated application of Nash-Williams’ Theorem 2.36 an infinite 푁 ⊆ 푀 such that for all 푆 ∈ 퐅, 푆|푁 = 푆 ∩ 퐹 |푁 is either empty or equal to 퐹 |푁. In other terms, for 퐅 a finite family of subsets of a front 퐹 on 푀, there exists 푁 ∈ [푀]∞ such that 퐅|푁 ⊆ {∅, 퐹 |푁}. Of course such a conclusion cannot be expected for a countably infinite family 퐅, as one easily see by considering for example the countably infinite family 퐅 = {{푠} ∣ 푠 ∈ 퐹 } of subsets of a non-trivial front 퐹 . However we obtained the following which is the main combinatorial result in this chapter. Theorem 3.24 (with R. Carroy). Let 퐹 be a front on 휔 and let 퐒 be a countable family of subsets of 퐹 . For all 푀 ∈ [휔]∞ there exists 푁 ∈ [푀]∞ such that 퐒|푁 consists of blocks of 퐹 |푁, i.e. 퐒|푁 ⊆ Blocks(퐹 |푁).

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Proof. By induction on the rank of 퐹 . For the trivial front, the theorem is trivial. Suppose that 퐹 is a front of strictly positive rank on 휔 and that the statement of the theorem holds for fronts of strictly smaller rank. Let 푀 ∈ [휔]∞. Claim. There exists 푋 ∈ [푀]∞ such that for all 푚 ∈ 푋 and all 푆 ∈ 퐒 we have (푆|푋)푚 ∈ Blocks ((퐹 |푋)푚).

Proof of the claim. For each 푛 ∈ 휔, since 퐹푛 is a front of strictly smaller rank than 퐹 , we can apply our induction hypothesis to the countable family 퐒푛 = {푆푛 ∣ 푆 ∈ 퐒} of subsets of 퐹푛. We thus build recursively a sequence (푋푖)푖∈휔 of infinite subsets of 푀 with 푘푖 = min(푋푖), such that

∞ (i) 푋0 = 푀 and 푋푖+1 ∈ [푋푖/푘푖] for every 푖 ∈ 휔; (ii) for every 푖 ∈ 휔 we have 퐒 |푋 ⊆ Blocks (퐹 |푋 ). 푘푖 푖+1 푘푖 푖+1

By induction, suppose that the sequence is defined up to 푖 ⩾ 0 and let 푘푖 = min 푋 . Then 퐹 is a front on 푀/푘 whose rank is strictly smaller than the 푖 푘푖 푖 rank of 퐹 . We consider the family 퐒 of subsets of 퐹 . By our induction hypo- 푘푖 푘푖 thesis there exists 푋 ∈ [푋 /푘 ]∞ such that 퐒 |푋 ⊆ Blocks (퐹 |푋 ). 푖+1 푖 푖 푘푖 푖+1 푘푖 푖+1 We then set 푋 = {푘푖 ∣ 푖 ∈ 휔}. To see that 푋 satisfies the Claim, let 푘푖 ∈ 푋 and 푆 ∈ 퐒. Since 푋/푘푖 ⊆ 푋푖+1 we have

(푆|푋) = 푆 |푋 = 푆 |푋/푘 = 푆 |푋 |푋/푘 ∈ Blocks (퐹 |푋 )|푋/푘 , 푘푖 푘푖 푘푖 푖 푘푖 푖+1 푖 푘푖 푖+1 푖 by the choice of 푋. Therefore by Lemma 3.23, it follows that

(푆|푋) ∈ Blocks (퐹 |푋/푘 ) = Blocks ((퐹 |푋) ), 푘푖 푘푖 푖 푘푖 as desired. By the Claim, there is no loss of generality in assuming that 퐹 is a front on 푀 and that for all 푚 ∈ 푀 and all 푆 ∈ 퐒 we have 푆푚 ∈ Blocks(퐹푚). 푖 We now fix an enumeration {푆 ∣ 푖 ∈ 휔} of 퐒. We build a sequence (푁푖)푖∈휔 of infinite subsets of 푀 with 푛푖 = min 푁푖 such that

∞ (i) 푁0 = 푀 and 푁푖+1 ∈ [푁푖/푛푖] ;

푗 푗 (ii) for every 푖 ∈ 휔 and every 푗 < 푖, either 푆 |푁푖 is empty, or 푆 |푁푖 = 퐹 |푁푖.

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By induction suppose that the sequence is defined up to 푖 ⩾ 0. Applying Nash- ∞ Williams’ Theorem 푖 + 1 many times, there exists 푁푖+1 ∈ [푁푖/푛푖] such that 푗 for every 푗 < 푖 + 1 either 푆 |푁푖+1 is empty or equal to 퐹 |푁푖+1. Now for 푁 = {푛0, 푛1, …}, we claim that for all 푆 ∈ 퐒 we have 푆|푁 is a block of 퐹 |푁. To see this let 푆푗 ∈ 퐒. Firstly, for every 푖 we have

푗 푗 (푆 |푁) = 푆푛 |푁 ∈ Blocks (퐹 )|푁 = Blocks (퐹 |푁) ), 푛푖 푖 푛푖 푛푖 by Lemma 3.23. Secondly, since 푁/푘푗 ⊆ 푁푗+1 we have

푗 푗 푆 |(푁/푘푗) = (푆 |푁푗+1)|푁/푘푗

Theorem 3.25 (with R. Carroy). Let 퐹 be a front on some 푀 ∈ [휔]∞. For all 푓 ∶ 퐹 → 2휔 there exists an infinite 푁 ⊆ 푀 such that the restriction 푓 ∶ 퐹 |푁 → 2휔 is uniformly continuous.

Proof. Applying Theorem 3.24 to 퐒 = {푓 −1(퐶) ∣ 퐶 is clopen in 2휔} we obtain an infinite 푁 ⊆ 휔 such that the restriction 푓|푁 ∶ 퐹 |푁 → 푋 satisfies that for all clopen set 퐶 we have

(푓|푁)−1(퐶) = 푓−1(퐶) ∩ 퐹 |푁 = 푓−1(퐶)|푁 ∈ Blocks(퐹 |푁).

Therefore 푓|푁 is uniformly continuous by Proposition 3.13. This result generalises to arbitrary compact metric space using the fact that every such space is a continuous image of the Cantor space as we already noticed in Theorem 3.8.

Theorem 3.26. Let 퐹 be a front on some 푀 ∈ [휔]∞ and 풳 be a compact metric space. For all 푓 ∶ 퐵 → 풳 there exists an infinite 푁 ⊆ 푀 such that the restriction 푓 ∶ 퐵|푁 → 풳 is uniformly continuous.

Proof. If 풳 is a non-empty compact metric space, then there exists a continuous ℎ ∶ 2휔 → 풳 onto 풳 [Kec95, (4.18), p. 23], and this ℎ is in fact uniformly continuous by the Heine-Cantor theorem [Kec95, (4.5), p. 19]. So let 푓 ∶ 퐹 → 풳 be any super-sequence. We can choose some super-sequence

65 3 Sequences in spaces

푔 ∶ 퐹 → 2휔 such that ℎ ∘ 푔 = 푓. Now applying Theorem 3.25 to 푔, there exists ′ ′ 휔 some front 퐹 ⊆ 퐹 such that 푔↾퐹 ′ ∶ 퐹 → 2 is uniformly continuous. It fol- ′ lows that 푓↾퐹 ′ = ℎ ∘ 푔↾퐹 ′ ∶ 퐹 → 풳 is uniformly continuous as a composition of uniformly continuous maps. In Subsection 2.2.3 we considered [푋]∞ for 푋 ∈ [휔]∞ with the induced topology from 2휔. We noticed it was homeomorphic to the Baire space and we viewed it as the closed subset of 휔휔 consisting of strictly increasing sequences in 푋. However as a subspace of 2휔, the set [푋]∞ also receives a unique uniformity. As a uniform space [푋]∞ is not complete, its completion is of course the Cantor space 2푋. The Boolean algebra of blocks of [푋]∞ consists of finite union of sets of the form

푁(퐹 , 퐺) = {푌 ∈ [푋]∞ ∣ 퐹 ⊆ 푌 and 퐺 ∩ 푌 = ∅}, for 퐹 , 퐺 ∈ [푋]<∞.

We have an analogue to Theorem 3.8 in the context of locally constant maps from [휔]∞ into a compact metric space. Theorem 3.27. Let 풳 be compact metrisable and ℎ ∶ [휔]∞ → 풳 be locally constant. There exists 푌 ∈ [휔]∞ such that the restriction ℎ ∶ [푌 ]∞ → 풳 is uniformly continuous, and therefore extends to a continuous map ℎ ∶ 2푌 → 풳. Proof. By Theorem 3.26 there exists some 푌 ∈ [휔]∞ such that the restriction ℎ̌ ∶ 퐹 ℎ|푌 → 풳 is uniformly continuous. This 푌 is as required since ℎ ∶ [푌 ]∞ → 풳 extends to the continuous ℎ ∶ 2푌 → 풳 given by ℎ(푠) = ℎ(푠)̌ if 푠 ∈ 퐹 ℎ|푌 and ℎ(푢) = ℎ(푠)̌ for the unique 푠 ∈ 퐹 ℎ|푌 with 푠 ⊏ 푢, otherwise.

3.4 Converging super-sequences

One can think of a super-sequence 푓 ∶ 퐹 → 풳 which extends to a continuous 푓 ∶ 퐹 → 풳 as ‘converging towards 푓’. In this section, we study these ‘converging super-sequences’, namely the continuous maps 푓 ∶ 퐹 → 풳 into some topological space 풳.

3.4.1 Normal form Let 푓 ∶ 퐹 → 풳 be a map from the closure of a front 퐹 on some 푁 ∈ [휔]∞ into a topological space 풳. Let

Λ푓 = {푠 ∈ 퐹 ∣ 푓(푠) is limit in 풳},

66 3 Sequences in spaces the closed subset of 퐹 of those points whose image is a limit point in 풳. ∞ Observe that if 푀 ∈ [푁] and 푔 = 푓↾퐹|푀 ∶ 퐹 |푀 → 풳, then Λ푔 = Λ푓 |푀. Recall that by Corollary 3.7, for all 푀 ∈ [푁]∞ we have 퐹 |푀 = 퐹 |푀 and (퐹 ) = 퐹 for all 푛 ∈ 푁. 푛 푛 Definition 3.28. Let 퐹 be a front on 푁 ∈ [휔]∞ and 풳 a topological space. We say that a continuous map 푓 ∶ 퐹 → 풳 is normal if

(1) Λ푓 is either empty or the closure of a front on 푋,

(2) for all 푠, 푡 ∈ 퐹 ∖ Λ푓 , 푠 ⊑ 푡 implies 푓(푠) = 푓(푡). Example 3.29. The map 푓 ∶ [휔]⩽2 → [휔]⩽2 given by

0휔 if 푚 = 0 and 푛 = 1, 푓(푚, 푛) = { 0푚10푛10휔 otherwise,

푚 휔 휔 푓(푚) = 0 10 and 푓(∅) = 0 , is continuous but it is not normal since Λ푓 = [휔]⩽1 ∪ {(0, 1)} is not the closure of a front.

Lemma 3.30. If 푓 ∶ 퐹 → 풳 is normal and 퐺 ⊆ 퐹 is a front then 푔 = 푓↾퐺 ∶ 퐺 → 풳 is normal too.

∞ Proof. Let 푀 ∈ [⋃ 퐹 ] with 퐺 = 퐹 |푀. Then Λ푔 = Λ푓 |푀 is either empty or a front on 푀. Moreover if 푠, 푡 ∈ 퐺∖Λ푔, then 푠, 푡 ∈ 퐹 , 푠, 푡 ⊂ 푀, and 푠, 푡 ∉ Λ푓 |푀, so 푠, 푡 ∉ Λ푓 and thus 푠 ⊑ 푡 implies 푔(푠) = 푓(푠) = 푓(푡) = 푔(푡). Here is Theorem 1.2 from the Introduction. Theorem 3.31 (with R. Carroy). Let 퐹 be a front on 푁 ∈ [휔]∞, 풳 be a topological space and 푓 ∶ 퐹 → 풳 be a continuous map. Then there exists a front 퐺 ⊆ 퐹 such that 푔 = 푓↾퐺 ∶ 퐺 → 풳 is normal. Proof. By induction on the rank of 퐹 . The theorem is obvious for the trivial front. We suppose it holds for all continuous maps from the closure of a front with rank strictly smaller than 훼 > 0. Let 퐹 be a front on 푁 with rk 퐹 = 훼. Claim. There exists 푀 ∈ [푁]∞ such that for all 푚 ∈ 푀 the map

푓푀 ∶ (퐹 |푀) ⟶ 풳 푚 푚 푠 ⟼ 푓({푚} ∪ 푠) is normal.

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Proof of the claim. We build by recursion a sequence (푀푖)푖∈휔, with 푚푖 the minimum of 푀푖, such that for all 푖 ∈ 휔 we have

∞ (a) 푀0 = 푁 and 푀푖+1 ∈ [푀푖/푚푖] ,

푀푖 (b) 푓푚푖 ∶ (퐹 |푀푖+1) → 풳 is normal. 푚푖 By induction, suppose that the sequence is defined up to 푖 ⩾ 0. The family 퐹 is a front on 푀 /푚 with rk(퐹 ) < 훼, so we can use the induction 푚푖 푖 푖 푚푖 hypothesis on the continuous map 푓 ∶ 퐹 → 풳 defined by 푠 ↦ 푓({푚 } ∪ 푠) 푚푖 푚푖 푖 ∞ 푀푖+1 to get 푀푖+1 ∈ [푀푖/푚푖] such that the restriction 푓푚푖 ∶ (퐹 |푀푖+1) → 풳 푚푖 is normal. ∞ 푀/푚푖 Now setting 푀 = {푚0, 푚1, …} we have 푀/푚푖 ∈ [푀푖+1] and thus 푓푚푖 is normal by Lemma 3.30. Therefore we can suppose without loss of generality that 퐹 is a non-trivial front on 푁 and that 푓 ∶ 퐹 → 풳 is such that

푓푛 ∶ 퐹 푛 ⟶ 풳 푠 ⟼ 푓({푛} ∪ 푠) is normal for all 푛 ∈ 푁. Notice that as 퐹 is a non-trivial front on 푁, {푛} ∈ 퐹 for all 푛 ∈ 푁. We now distinguish two cases.

∅ ∉ Λ푓 : Since 푓 is continuous we have 푓(∅) = lim푛∈푁 푓({푛}) and as 푓(∅) is isolated in 풳 there exists a 푀 ∈ [푁]∞ such that 푓({푚}) = 푓(∅) for all 푚 ∈ 푀. Then for all 푚 ∈ 푀 we have that {푚} ∉ Λ푓 , that is ∅ ∉ (Λ푓 ){푚}. Hence (Λ푓 ){푚} cannot be the closure of a front, so it must be empty. Therefore Λ푓 |푀 is empty and for 퐺 = 퐹 |푀, the restriction

푓↾퐺 ∶ 퐺 → 풳 is constant, hence normal. ∞ ∅ ∈ Λ푓 : There exists 푀 ∈ [푁] such that either {푚} ∉ Λ푓 for all 푚 ∈ 푀, or {푚} ∈ Λ푓 for all 푚 ∈ 푀.

– In the former case, we have Λ푓 |푀 = {∅} and so 퐺 = 퐹 |푀 meets the conditions.

– In the latter case, for all 푚 ∈ 푀 the set (Λ푓 |푀)푚 is the closure of a front 퐿푚 on 푀/푚. Hence 퐿 = seq푚∈푀 퐿푚 is a front on 푀 and Λ푓 |푀 = 퐿. Hence 퐺 = 퐹 |푀 meets the requirements.

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Remark 3.32. It does not hold that for every map 푓 ∶ 퐹 → 2 from the closure −1 of a front there exists a front 퐺 ⊆ 퐹 such that at least one of 푓↾퐺 (0) or −1 푓↾퐺 (1) is either empty or the closure of a front. One can for example consider the function 푓 ∶ [휔]⩽2 → 2 defined by

0 if 푚 is even, 1 if 푚 is odd, 푓(푚, 푛) = { 푓(푚) = { 푓(∅) = 0. 1 if 푚 is odd, 0 if 푚 is even,

3.4.2 Super-sequences of isolated points In Section 4.2 we deal with super-sequences 푓 ∶ 퐹 → 풳 which range in Isol(풳), the isolated points of 풳, and which continuously extend to a map 푓̄ ∶ 퐹 → 풳.

In this case, the set Λ푓 ̄ is then included in 퐹 ∖ 퐹 . We collect here some easy lemmas pertaining to this situation.

Lemma 3.33. If 퐹 is a front then 퐹 ∖ 퐹 = {휒푠 ∣ ∃푡 ∈ 퐹 푠 ⊏ 푡} and 퐹 = max⊑ 퐹 .

Proof. By Proposition 3.5 we have {휒푠 ∣ ∃푡 ∈ 퐹 푠 ⊑ 푡} = 퐹 . For the first claim, if 푠 ⊑ 푡 ∈ 퐹 and 푠 ∉ 퐹 then 푠 ⊏ 푡. Conversely, if 푠 ⊏ 푡 ∈ 퐹 then 푠 ∉ 퐹 since 퐹 is a front. For the second claim, if 푠 ∈ 퐹 and 푠 ⊑ 푡 ∈ 퐹 , then there is 푢 ∈ 퐹 with 푡 ⊑ 푢, so 푠 ⊑ 푢 and thus 푠 = 푡 = 푢. And if 푠 ∈ 퐹 is maximal in 퐹 then there is no 푡 ∈ 퐹 with 푠 ⊏ 푡 and thus by the first claim 푠 ∈ 퐹 .

Lemma 3.34. If 퐹 and 퐺 are fronts and 퐺 ⊆ 퐹 ∖ 퐹 then 1 + rk 퐺 ⩽ rk 퐹 .

Proof. Notice that 푇 (퐺) ⊆ 푇 (퐹 ). By induction we show that for every 푠 ∈ 푇 (퐺) we have 1 + 휌푇 (퐺)(푠) ⩽ 휌푇 (퐹)(푠). If 푠 ∈ 퐺 then there exists 푡 ∈ 퐹 with 푠 ⊏ 푡 and so 1 + 휌푇 (퐺)(푠) = 1 ⩽ sup{휌푇 (퐹)(푡) + 1 ∣ 푠 ⊏ 푡 ∈ 푇 (퐹 )} = 휌푇 (퐹)(푠). Now if 푠 ∈ 푇 (퐺) ∖ 퐺 then by induction hypothesis

1 + 휌푇 (퐺)(푠) = sup{(1 + 휌푇 (퐺)(푡)) + 1 ∣ 푠 ⊏ 푡 ∈ 푇 (퐺)}

⩽ sup{휌푇 (퐹)(푡) + 1 ∣ 푠 ⊏ 푡 ∈ 푇 (퐹 )} = 휌푇 (퐹)(푠).

Therefore 1 + rk 퐺 = 1 + 휌푇 (퐺)(∅) ⩽ 휌푇 (퐹)(∅) = rk 퐹 . Remark 3.35. The topology induced by the Cantor space on every front 퐹 is discrete. It easily follows from this fact that the set of isolated points Isol(퐹 ) of the closure of 퐹 is equal to 퐹 , or in other words the Cantor-Bendixson derivative of 퐹 is equal to 퐹 . However, in general it is not true that 퐹 ∖ 퐹 is the closure of a front as the following example shows.

69 3 Sequences in spaces

⋯ ⋯

Bendixson The closed set 퐺 ...and its derivative

Figure 3.3: A front whose derivative of the closure is no closure of a front.

Let 퐺(0) = {{1}} ∪ [휔/1]2 and 퐺(푛) = [휔/푛]2 for every 푛 ⩾ 1. We build the front 퐺 = seq푛∈휔 퐺(푛) (cf. Figure 3.3). We have

퐺′ = Isol (퐺 ∖ 퐺) = {{0, 푛} ∣ 푛 ⩾ 2} ∪ [휔/1]2.

But while ⋃ 퐺′ = 휔, the infinite set 휔 has no initial segment in 퐺′. Hence 퐺′ is not a front.

Lemma 3.36. Let 퐹 be a front, 풳 a topological space. If 푓 ∶ 퐹 → Isol(풳) is spare and extends to a normal 푓 ∶ 퐹 → 풳, then Λ푓 = 퐹 ∖ 퐹 .

Proof. Since 푓 ∶ 퐹 → Isol(풳) we have Λ푓 ⊆ 퐹 ∖퐹 by definition. Let 푠 ∈ 퐹 ∖Λ푓 . Since 푓 ∶ 퐹 → 풳 is normal then for every 푡 ∈ 퐹 we have 푠 ⊑ 푡 implies 푓(푠) = 푓(푡). Since 푓 ∶ 퐹 → 풳 is spare, necessarily 푠 ∈ 퐹 by Lemma 2.45. In general for a front 퐹 , 퐹 ∖ 퐹 , which is the Cantor-Bendixson derivative of 퐹 , is not of the form 퐺 for a front 퐺. However when this happens, as in the previous lemma, we have the following result.

Lemma 3.37. Let 퐹 be a front on 푋. If 퐹 ∖ 퐹 = 퐺 for some front 퐺 on 푋, then for every 푠 ∈ 퐺 the following are equivalent

(i) 푠 ∈ 퐺,

(ii) for every 푛 ∈ 푋/푠, 푠 ∪ {푛} ∈ 퐹 ,

(iii) there exists 푛 ∈ 푋/푠 such that 푠 ∪ {푛} ∈ 퐹 .

And moreover, 퐹 = {푠 ∪ {푛} ∣ 푠 ∈ 퐺 and 푛 ∈ 푋/푠}.

70 3 Sequences in spaces

Proof. (i)→(ii): Let 푛 ∈ 푋/푠. Since 퐹 is a front on 푋, there exists 푡 ∈ 퐹 with 푡 ⊏ 푠 ∪ {푛} ∪ 푋/푛. Now 푡 ⊑ 푠 would contradict Lemma 3.33 for 퐹 . And if 푠 ∪ {푛} ⊏ 푡 then 푠 ∪ {푛} ∈ 퐹 ∖ 퐹 = 퐺 by Lemma 3.33 and so 푠 ⊏ 푠 ∪ {푛} ∈ 퐺 contradicting Lemma 3.33 for 퐺. So 푠 ∪ {푛} = 푡 ∈ 퐹 as desired.

(iii)→(i): If for some 푛 ∈ 푋/푠 we have 푠 ∪ {푛} ∈ 퐹 , then by Lemma 3.33 푠 is maximal for ⊑ in 퐺 and so 푠 ∈ 퐺. Finally by (ii) 푠 ∪ {푛} ∈ 퐹 for every 푠 ∈ 퐺 and 푛 ∈ 푋/푠. Conversely if 푠 ∪ {푛} ∈ 퐹 then 푠 ∈ 퐺 and thus 푠 ∈ 퐺 by (iii). This concludes this chapter on super-sequences in metric and topological spaces. The results we obtained are applied in the context of bqo theory in the next chapter.

71 4 The ideal space of a well-quasi-order

For simplicity we restrict ourselves to partial orders in this chapter. This is of course no real restriction since we are only interested in the properties of quasi-orders preserved by equivalence and every quasi-order admits a unique equivalent partial order up to isomorphism. The following ‘ideal objects’ of a partial ordered play a central rôle in this chapter. Definition 4.1. Let 푃 be a partial order. An ideal (also directed ideal, up- ideal) of 푃 is a subset 퐼 ⊆ 푃 such that (1) 퐼 is non empty;

(2) 퐼 is a downset;

(3) for every 푝, 푞 ∈ 퐼 there exists 푟 ∈ 퐼 with 푝 ⩽ 푟 and 푞 ⩽ 푟. Equivalently a subset 퐼 of a po 푃 is an ideal if 퐼 a downset and 퐼 is directed, namely every (possibly empty) finite subset 퐹 ⊆ 퐼 admits an upper bound in 퐼, i.e. there exists 푞 ∈ 퐼 with 퐹 ⊆ ↓ 푞. For every 푝 ∈ 푃 , the set ↓ 푝 is an ideal called a principal ideal. Definition 4.2. The set of ideals of 푃 partially ordered by inclusion, denoted by Id(푃 ), is referred to as the ideal completion of the partial order 푃 . For each partial order 푃 we have the embedding

푃 ⟼ Id(푃 ) 푝 ⟼ ↓ 푝, and we henceforth identify each element 푝 with the corresponding principal ideal ↓ 푝. In particular we have the inclusions 푃 ⊆ Id(푃 ) ⊆ 풟(푃 ) as partial orders. We observe that we cannot replace 풟(푃 ) by Id(푃 ) in Proposition 2.6, i.e. it is not true that a po 푃 is wqo if and only Id(푃 ) is well-founded. The simplest

72 4 The ideal space of a well-quasi-order example is given by the antichain 퐴 = (휔, =). The partial order Id(퐴) is equal to 퐴 so, in particular, even though 퐴 is not wqo, Id(퐴) is well-founded. Nonetheless, when 푃 is wqo then Id(푃 ) is well-founded. Recall also that by Corollary 2.56 if 푃 is bqo, then 푉 ∗(푃 ) is bqo, and so in particular Id(푃 ) – which is included in 푉 ∗(푃 ) – is bqo. Of course since 푃 ⊆ Id(푃 ), it follows that 푃 is bqo if and only if Id(푃 ) is bqo. A non-principal ideal of 푃 is an ideal which is not of the form ↓ 푝 for some 푝 ∈ 푃 . We write Id*(푃 ) for the partial order of non-principal ideals, i.e. Id*(푃 ) = Id(푃 ) ∖ 푃 . The partial order Id*(푃 ) is therefore the remainder of the ideal completion of 푃 . The main new result in this chapter is a proof of the following conjecture due to Pouzet [Pou78]: If 푃 is wqo and Id*(푃 ) is bqo, then 푃 is bqo. Notice that the assumption that 푃 is wqo cannot be dropped since for both the antichain (휔, =) and the descending chain 휔op – the opposite of (휔, ⩽) – the remainder of the ideal completion is empty, hence bqo. Our proof of Pouzet’s conjecture relies on the fact that the ideal completion Id(푃 ) of a wqo 푃 is actually a compact topological space when topologised as a subspace of the generalised Cantor space 2푃 and that the results we obtained in Chapter 3 can eventually be applied to yield the following (cf. Theorem 4.34):

Theorem 4.3. Every super-sequence 푓 ∶ 퐹 → 푃 into a wqo 푃 admits a Cauchy sub-super-sequence 푓′ ∶ 퐹 ′ → 푃 which therefore extends to a continuous map 푓′ ∶ 퐹 ′ → Id(푃 ) into the ideals of 푃 .

Using the above theorem and the results of Chapter 3 we then prove (cf. Theorem 4.39) that any bad super-sequence in a wqo 푃 yields a bad super- sequence into the non principal ideals of 푃 . This shows that if a wqo 푃 is not bqo, then Id*(푃 ) is not bqo, therefore proving Pouzet’s conjecture. Our first task is to collect the properties of the ideal completion ofa wqo which are required to prove the above theorem. We do this in the first section by showing that in the case of a wqo the ideal completion coincides with two other important completions of a quasi-order – namely the Cauchy ideal completion and the profinite completion or Nachbin order-compactification – the properties of which combine to yield what we call the ideal space of a wqo.

73 4 The ideal space of a well-quasi-order

4.1 Completions of well-quasi-orders

4.1.1 Ideal completion Let us start by the well-known fact that ideals play a particular rôle among the downsets of a partial order.

Proposition 4.4. For a non empty downset 퐼 of a partial order 푃 , the following are equivalent.

(i) 퐼 is an ideal;

(ii) 퐼 is a join-prime element of the lattice 풟(푃 ), i.e. for every 퐸, 퐷 ∈ 풟(푃 ), 퐼 ⊆ 퐸 ∪ 퐷 implies 퐼 ⊆ 퐸 or 퐼 ⊆ 퐷;

(iii) for every 퐸, 퐷 ∈ 풟fb(푃 ), 퐼 ⊆ 퐸 ∪ 퐷 implies 퐼 ⊆ 퐸 or 퐼 ⊆ 퐷. Proof. (i)→(ii): For an ideal 퐼 and downsets 퐷 and 퐸, suppose that 퐼 ⊈ 퐷 and 퐼 ⊈ 퐸. Then there exist 푝, 푞 ∈ 퐼 with 푝 ∉ 퐷 and 푞 ∉ 퐸, and since 퐼 is directed there is some 푟 ∈ 퐼 with 푟 ⩾ 푝 and 푟 ⩾ 푞. As 퐸 and 퐷 are downsets, necessarily 푟 does not belong to 퐷 nor to 퐸. Therefore 퐼 ⊈ 퐸 ∪ 퐷.

(ii)→(iii): Trivial.

(iii)→(i): Suppose 퐼 is a non empty downset which is not directed. Then there exist 푝, 푞 ∈ 퐼 such that ↑ 푝∩↑ 푞∩퐼 is empty. Therefore 퐼 ⊆푃∖↑푝∪푃∖↑푞 while both 퐼 ⊈ 푃 ∖ ↑ 푝 and 퐼 ⊈ 푃 ∖ ↑ 푞 hold. A partial order in which every directed set 퐷 admits a supremum, denoted by sup 퐷, is called a directed complete partial order (dcpo, also up-complete). Since any directed union of ideals of a po 푃 is an ideal of 푃 , Id(푃 ) is a dcpo. Recall that an element 푝 of dcpo is compact if for every directed subset 퐷, 푝 ⩽ sup 퐷 implies that there exists 푞 ∈ 푆 with 푝 ⩽ 푞. Finally a partial order 푃 is said to be algebraic (also algebraic domain) if it is a dcpo and for each 푝 ∈ 푃 there exists a directed set 퐷 of compact elements with sup 퐷 = 푝. It is easy to see that Id(푃 ) is an algebraic partial order whose compact elements are exactly the principal ideals. Therefore every partial order is isomorphic to the partial order of compact elements of some algebraic partial order. In fact, conversely a partial order is algebraic if and only if it is isomorphic to the ideal completion of some partial order (see for example [Ern93]). We also refer the reader to the paper by Gehrke and Priestley [GP08, Proposition 2.1] for an abstract characterisation of the ideal completion of a partial order.

74 4 The ideal space of a well-quasi-order

We now endow the ideal completion of a partial order with a topology. We do this by exploiting the fact that any dcpo can be equipped with several intrinsic topologies, i.e. topologies which can be defined by in terms of the partial order. We recall the relevant definitions and refer the reader to the textbook by Gierz et al. [Gie+03] for a more detailed presentation of these concepts. For a dcpo 푃 , a subset 푈 of 푃 is called Scott open if it is an upset and for every directed 푆, sup 푆 ∈ 푈 implies 푆 ∩ 푈 ≠ ∅. Scott open subsets of a dcpo 푃 form a topology on 푃 called the Scott topology. The lower topology on a partial order 푃 is generated by the base consisting of the finitely bounded downsets, namely the subsets of the form 푃 ∖ ↑ 퐹 for finite 퐹 ⊆ 푃 . The Lawson topology on a dcpo 푃 is the join of the Scott and the lower topology, it admits as a subbase the family of Scott open sets and sets of the form 푃 ∖ ↑ 푝, for 푝 ∈ 푃 , and as a base the sets of the form 푈 ∖ ↑ 퐹 for 푈 Scott open and 퐹 a finite subset of 푃 . For every partial order 푃 let us endow Id(푃 ) with the Lawson topology. It turns out that this topology on Id(푃 ) admits another useful description. Recall from Subsection 2.1.4 that the clopen subsets of the generalised Cantor space 2푃 are finite unions of sets of the form

푁(퐹 , 퐺) = {푋 ⊆ 푃 ∣ 퐹 ⊆ 푋 and 퐺 ∩ 푋 = ∅} for finite subsets 퐹 and 퐺 of 푃 . We write ⟨퐹 ⟩ instead of 푁(퐹 , ∅) and ⟨푝⟩ instead of ⟨{푝}⟩. In particular, the sets of the form ⟨푝⟩ or ⟨푝⟩∁ for 푝 ∈ 푃 constitute a subbase for the topology of 2푃 . As a matter of fact, the topology on 2푃 is the Lawson topology associated with the partial order of inclusion on the subsets of 푃 . We now notice that the topology induced on Id(푃 ) by 2푃 is intrinsic:

Lemma 4.5. Let 푃 be a partial order. The Lawson topology on Id(푃 ) coincides with the topology induced by 2푃 .

Proof. For 푝 ∈ 푃 , ⟨푝⟩ ∩ Id(푃 ) is an upset for inclusion. In addition, if 푆 is a directed set of ideals and 푝 ∈ ⋃ 푆 then for some 퐼 ∈ 푆 we have 푝 ∈ 퐼, i.e. 퐼 ∈ ⟨푝⟩. Therefore ⟨푝⟩ is Scott open. Moreover if 푝 ∈ 푃 , then ⟨푝⟩∁ ∩ Id(푃 ) is equal to {퐼 ∈ Id(푃 ) ∣ ↓ 푝 ⊈ 퐼}, which is open in the lower topology by definition. Conversely, let 푈 be a Scott open set of Id(푃 ). For every 퐼 ∈ 푈 since 퐼 is the directed union of the principal ideals ↓ 푝 for 푝 ∈ 퐼, there must exist 푝 ∈ 퐼 with ↓ 푝 ∈ 푈, and therefore 퐼 ∈ ⟨푝⟩ ∩ Id(푃 ) ⊆ 푈. Therefore 푈 is open in the relative topology. Moreover for 퐼 ∈ Id(푃 ), the lower open subbasic set

75 4 The ideal space of a well-quasi-order

{퐽 ∈ Id(푃 ) ∣ 퐼 ⊈ 퐽} is equal to the set Id(푃 ) ∩ ⋃ ⟨푝⟩∁ which is open in the 푝∈퐼 relative topology. In particular, since a subspace of the compact Hausdorff space 2푃 is compact if and only if it is closed, we have the following.

Corollary 4.6. The partial order Id(푃 ) is compact in the Lawson topology if and only if Id(푃 ) is closed as a subset of 2푃 .

Remark 4.7. Any partial order 푃 can be equipped with the Alexandroff topology consisting in the upsets of 푃 . The condition (ii) in Proposition 4.4 amounts to saying that the ideals are the irreducible non empty closed sets of 푃 for the Alexandroff topology. Along this line, Hoffmann [Hof79] observed that the sobrification of a partial order 푃 equipped with the Alexandroff topology coincide with the partial order Id(푃 ) equipped with the Scott topology (see for example [Joh86, p. 291] for a proof).

4.1.2 The Cauchy ideal completion We now regard a partial order 푃 as a subset of the generalised Cantor space 2푃 via the identification 푝 ↦ ↓ 푝.

Definition 4.8. The topological closure of 푃 inside 2푃 equipped with the partial order of inclusion is denoted by CId(푃 ) and referred to as the Cauchy ideal completion of the partial order 푃 .

Erné and Palko [EP98] call a subset 퐽 of a partial order 푃 a Cauchy ideal if for every finite 퐹 ⊆ 퐽 and every finite 퐺 ⊆ 푃 ∖ 퐽 there exists 푝 ∈ 푃 with 퐹 ⊆ ↓ 푝 and 퐺 ∩ ↓ 푝 = ∅. In our terminology, a subset 퐽 ⊆ 푃 is a Cauchy ideal if and only if for every finite 퐹 , 퐺 ⊆ 푃 with 퐽 ∈ 푁(퐹 , 퐺) there exists 푝 ∈ 푃 such that ↓ 푝 ∈ 푁(퐹 , 퐺). Whence in other words a subset 퐽 of 푃 is a Cauchy ideal exactly when 퐽 ∈ CId 푃 . The term ‘Cauchy ideal completion’ is introduced by Erné and Palko [EP98] who define on any partial order 푃 a uniformity 푈푃 by taking as subbasic entourages the sets of the form

푈↑ 푝 = {(푟, 푠) ∈ 푃 × 푃 ∣ 푝 ⩽ 푟 ↔ 푝 ⩽ 푠} = (↑ 푝 × ↑ 푝) ∪ (푃 ∖ ↑ 푝 × 푃 ∖ ↑ 푝), for 푝 ∈ 푃 . Now by the results of Section 3.2, the uniformity induced by the unique compatible uniformity on the compact Hausdorff space 2푃 is described via the blocks of 푃 .

76 4 The ideal space of a well-quasi-order

Lemma 4.9. Let 푃 be a partial order viewed as a subset of 2푃 via 푝 ↦ ↓ 푝. The family of blocks of 푃 is the Boolean algebra generated by the sets of the form ↑ 푝 for 푝 ∈ 푃 .

Proof. The family of clopen subsets of 2푃 being the Boolean algebra generated by the sets of the form ⟨푝⟩ for 푝 ∈ 푃 , it follows that the family of blocks of 푃 is the Boolean algebra of subsets of 푃 generated by the sets of the form ⟨푝⟩ ∩ 푃 for each 푝 ∈ 푃 . Via the identification 푝 ↦ ↓ 푝 we have

⟨푝⟩ ∩ 푃 = {푞 ∈ 푃 ∣ ↓ 푞 ∈ ⟨푝⟩} = {푞 ∈ 푃 ∣ 푝 ∈ ↓ 푞} = ↑ 푝.

Therefore Blocks(푃 ) is the Boolean algebra generated by the sets of the form ↑ 푝 for 푝 ∈ 푃 . It follows that the uniformity associated with Blocks(푃 ) (see Lemma 3.17) coincides with the uniformity 푈푃 . In particular, it follows that the uniform completion of (푃 , 푈푃 ) is (uniformly isomorphic to) the topological closure of 푃 in 2푃 . Therefore as observed by Erné and Palko [EP98] the Cauchy ideals may also be viewed as the points of the uniform completion of (푃 , 푈푃 ). This justifies our use of their terminology. We also refer the reader to the paper by Erné [Ern01] where the idea of uniform completion and quasi-uniform completion of partial orders is further explored. The following simple observation relates the ideal completion of a partial order with its Cauchy ideal completion.

Lemma 4.10. Let 푃 be a partial order.

(i) The set 풟(푃 ) is closed in 2푃 .

(ii) Every ideal is a Cauchy ideal.

In particular, we have the following inclusions of partial orders:

푃 ⊆ Id(푃 ) ⊆ CId(푃 ) ⊆ 풟(푃 ).

Proof. (i) As we already observed in Subsection 2.1.4, the set 풟(푃 ) is closed in 2푃 since 풟(푃 ) = ⋂⟨푞⟩∁ ∪ ⟨푝⟩. 푝⩽푞

(ii) Let 퐼 be an ideal of 푃 and 푁(퐹 , 퐺) – 퐹 , 퐺 ∈ [푄]<∞ – be any basic neighbourhood of 퐼 in 2푃 . Since 퐼 is directed, 퐹 ⊆ 퐼 implies that there exists 푝 ∈ 퐼 with 퐹 ⊆ ↓ 푝. Since 퐼 is downward closed, 퐺 ∩ 퐼 = ∅ implies

77 4 The ideal space of a well-quasi-order

퐺 ∩ ↓ 푝 = ∅. Therefore ↓ 푝 belongs to the neighbourhood 푁(퐹 , 퐺). Since 푁(퐹 , 퐺) was arbitrary, it follows that 퐼 ∈ CId(푃 ). Finally since 풟(푃 ) is closed, we have CId(푃 ) ⊆ 풟(푃 ) and by (ii) we have Id(푃 ) ⊆ CId(푃 ). In general Cauchy ideals need not be directed (nor do they need to be non- empty), and CId(푃 ) need not coincide with Id(푃 ). Two particularly simple examples where Id(푃 ) is not equal to CId(푃 ) are given by the antichain 퐴 = (휔, =) and the descending chain 퐷 = 휔op – the opposite of (휔, ⩽). Indeed, while as we already noticed Id(퐴) = 퐴, the Cauchy ideal completion CId(퐴) is the antichain 퐴 = {{푛} ∣ 푛 ∈ 휔} with the addition of a minimal element ∅ which is also a limit point. Similarly we have Id(퐷) = 퐷, while CId(퐷) consists of 퐷 together with a minimal element which is also a limit point. We now turn to the characterisation of the partial orders for which the ideal completion coincide with the Cauchy ideal completion. For any subset 푆 of a partial order 푃 let

푆↑ = ⋂ ↑ 푝 = {푞 ∈ 푃 ∣ ∀푝 ∈ 푆 푝 ⩽ 푞}, 푝∈푆 the upset of upper bounds of 푆 in 푃 . Also, as we did for infinite subsets of the natural numbers, let us denote by [푃 ]<∞ the set of finite subsets of a set 푃 .

Definition 4.11. We say that a partial order 푃 has property 푀 if the set of upper bounds of any finite (possibly empty) subset of 푃 admits a finite basis. In symbols, for all 퐹 ∈ [푃 ]<∞ there exists 퐺 ∈ [푃 ]<∞ such that 퐹 ↑ = ↑ 퐺.

Equivalently 푃 has property 푀 if 푃 is finitely generated and for every 푝, 푞 ∈ 푃 the upset ↑ 푝 ∩ ↑ 푞 is finitely generated. Recall that by Proposition 2.6 (W4) a partial order 푃 is wqo if and only if every upset is finitely generated. In particular every wqo has property 푀. However adding a minimal element ⊥ to the antichain (휔, =) provides an example of a partial order with property 푀 which is of course not wqo. Notice however that while any subset of a wqo is wqo, the property 푀 is not hereditary. Indeed removing the minimal element from (휔, =)∪{⊥} results in the loss of the property 푀. Remark 4.12. Bekkali, Pouzet, and Zhani [BPZ07] call a partial order 푃 up- closed if ↑ 푝 ∩ ↑ 푞 is finitely generated for every 푝, 푞 ∈ 푃 . Their notion of up-closed differs from property 푀 only in that 푃 = ∅↑ need not be finitely generated.

78 4 The ideal space of a well-quasi-order

•

• ⋯ • • • • • ⋯

⋮

• • • •

Figure 4.1: Examples of partial orders without property 푀.

Now observe that by Lemma 4.10 we have Id(푃 ) = CId(푃 ) if and only if Id(푃 ) is closed in 2푃 . Moreover by Corollary 4.6 Id(푃 ) is closed in 2푃 if and only if Id(푃 ) is compact in the Lawson topology. Finally it is folklore that a partial order 푃 has property 푀 if and only if Id(푃 ) is compact in the Lawson topology (see e.g. [Gie+03, Corollary II-5.15, p.259 and Proposition I-4.10, p.118]). However to keep our exposition self-contained we provide the proof of the following proposition which can also be found in the paper by Bekkali, Pouzet, and Zhani [BPZ07].

Proposition 4.13. For a partial order 푃 , the following are equivalent.

(i) 푃 has property 푀;

(ii) CId(푃 ) = Id(푃 ) as subsets of 2푃 ;

(iii) Id(푃 ) is compact in the Lawson topology.

In particular, 푃 has property 푀 if and only if every Cauchy ideal of 푃 is an ideal.

Proof. (i)→(ii): Suppose 푃 has property 푀. For all 퐹 ∈ [푃 ]<∞ there exists 퐺 ∈ [푃 ]<∞ such that 퐹 ↑ = ↑ 퐺 and therefore

⋃ ⟨푟⟩ = ⋃ ⟨푟⟩ 푟∈퐹 ↑ 푟∈퐺

is clopen in 2푃 as a finite union of clopen sets. Since ∅↑ = 푃 is also finitely generated there exists a finite 퐵 ⊆ 푃 with 푃 = ↑ 퐵. Now we can

79 4 The ideal space of a well-quasi-order

see that

Id(푃 ) ={퐼 ∈ 풟(푃 ) ∣ ∀퐹 ∈ [푃 ]<∞(퐹 ⊆ 퐼 → ∃푟 ∈ 퐹 ↑ 푟 ∈ 퐼) ∧ 퐼 ≠ ∅} = 풟(푃 ) ∩ ⋂ (⟨퐹 ⟩∁ ∪ ⋃ ⟨푟⟩) ∩ ⋃ ⟨푝⟩. 퐹∈[푃 ]<∞ 푟∈퐹 ↑ 푝∈퐵

is closed in 2푃 . Since 푃 ⊆ Id(푃 ) ⊆ CId(푃 ) by Lemma 4.10, we have CId(푃 ) = Id(푃 ).

(ii)→(iii): By Lemma 4.10 again, we have CId(푃 ) = Id(푃 ) if and only if Id(푃 ) is closed in 2푃 . Moreover by Corollary 4.6, Id(푃 ) is closed in 2푃 if and only if Id(푃 ) is compact in the Lawson topology.

(iii)→(i): By Corollary 4.6, if Id(푃 ) is compact in the Lawson topology, then Id(푃 ) is closed in 2푃 and therefore compact. In particular, the empty set – which is not an ideal by definition – does not belong to the closure of 푃 and so ∅ admits a basic neighbourhood 푁(∅, 퐺) for a finite subset 퐺 of 푃 such that ↓ 푝 ∉ 푁(∅, 퐺) for all 푝 ∈ 푃 , i.e. ↑ 퐺 = 푃 , or in other symbols ∅↑ = ↑ 퐺. Now let 퐹 ⊆ 푃 be non empty and finite. If 퐹 ↑ is empty it is equal to ↑ ∅, so suppose 퐹 ↑ is non empty. Notice that the compact subspace ⟨퐹 ⟩ ∩ Id(푃 ) is equal to the union of the basic open sets ⟨푝⟩ for 푝 ∈ 퐹 ↑. Consequently there exists a finite 퐺 ⊆ 퐹 ↑ with ⟨퐹 ⟩ ∩ Id(푄) = ⋃ ⟨푞⟩ and it follows that 퐹 ↑ = ↑ 퐺, as desired. 푞∈퐺 In particular when 푃 is wqo, since 푃 has property 푀, then the ideal completion Id(푃 ) equipped with the Lawson topology coincides with the Cauchy ideal completion CId(푃 ).

Notation 4.14. When 푃 is wqo, we make a slight abuse of notation and denote by Id(푃 ) the Cauchy ideal completion of 푃 , or equivalently the ideal completion of 푃 endowed with the Lawson topology. In this case, we refer to Id(푃 ) as the ideal space of the wqo 푃 .

For 푃 wqo, the ideal space Id(푃 ) is a partially ordered compact topological space. It actually enjoys further properties as we now see. Recall that a point 푥 of a topological space 풳 is isolated in 풳 if the singleton {푥} is open. A limit point of a topological space 풳 is a point that is not isolated, i.e. for every neighbourhood 푈 of 푥 there is a point 푦 ∈ 푈 with 푦 ≠ 푥. A topological space with no isolated points is called perfect. On the other extreme, a topological space is called scattered if it admits no perfect subspace.

80 4 The ideal space of a well-quasi-order

Proposition 4.15 ([PS06]). Let 푃 be wqo. Then the space Id(푃 ) is scattered. Moreover, an ideal 퐼 is isolated in Id(푃 ) if and only if 퐼 is principal. Proof. We show that 풟(푃 ) is scattered as a subspace of 2푃 . Let 푋 ⊆ 풟(푃 ) be non empty. Since 푃 is wqo, then 풟(푃 ) is well-founded by Proposition 2.6 so there exists a ⊆-minimal element 퐷 in 푋. Now since 푃 has the finite basis property by Proposition 2.6 there exists a finite 퐹 ⊆ 푃 with ↑ 퐹 = 푃 ∖ 퐷. It follows that 푁(∅, 퐹 )∩푋 = {퐷}, whence 퐷 is isolated in 푋. Since the property of being scattered is hereditary, it follows that Id(푃 ) is scattered. Now let 퐼 ∈ Id(푃 ). If 퐼 is isolated, {퐼} is open in Id(푃 ), and since 푃 is dense in Id(푃 ), 퐼 = ↓ 푝 holds for some 푝 ∈ 푃 and so 퐼 is principal. Conversely for 푝 ∈ 푃 let 퐹 be a finite basis for 푃 ∖ ↓ 푝. We have 푁({푞}, 퐹 ) ∩ Id(푃 ) = {↓ 푝}, so ↓ 푃 is isolated in Id(푃 ). Moreover we have the following corollary of Lemma 2.12:

Proposition 4.16. Let 푃 be wqo. Every sequence (퐼푛)푛∈휔 in Id(푃 ) admits a subsequence (퐼 ) which converges to the ideal ⋃ 퐼 inside Id(푃 ). In 푗 푗∈푁 푗∈푁 푗 particular, every sequence (푝푛)푛∈휔 ⊆ 푃 admits a subsequence (푝푛)푛∈푁 which converges to the ideal ↓{푝푛 ∣ 푛 ∈ 푁} in Id(푃 ). Proof. As 푃 is wqo, Id(푃 ) is a closed subset of 풟(푄) and therefore the first statement directly follows from Lemma 2.12. The second statement already follows from the more particular result stated in Fact 2.11. To continue our study of the ideal space of a wqo it is instructive to explore the dual description of the Cauchy ideal completion CId(푃 ) of an arbitrary partial order 푃 . Importantly, this will lead us to recognise the ideal space of a wqo as the profinite completion or Nachbin compactification of this wqo.

4.1.3 Duality and Cauchy ideal completion Zero-dimensional compact Hausdorff topological spaces play a prominent rôle in this thesis. Thanks to the seminal work of Stone [Sto36] these spaces are known to be intimately related to Boolean algebras. Their ordered analogues are the totally order-disconnected compact topological spaces, important examples of which are given in our context by the Cauchy ideal completion of a partial order. As discovered by Priestley [Pri72], totally-order disconnected compact spaces also possess a dual, algebraic, life as bounded distributive lattices. We now briefly recall the basic facts about these dualities and introduce some notations and conventions. We refer the reader to the monograph by Davey and Priestley [DP02] for details and proofs.

81 4 The ideal space of a well-quasi-order

A topological space 풳 equipped with a partial order ⩽ is totally order- disconnected if given 푥 ⩽̸ 푦 in 풳, there exists a clopen upset 푈 such that 푥 ∈ 푈 and 푦 ∉ 푈.A Priestley space is a totally order-disconnected compact topological space. For every set 퐸 the generalised Cantor space 2퐸 partially ordered by set-inclusion ⊆ is a Priestley space. In fact up to isomorphism the Priestley spaces are exactly the closed subspaces of a space (2퐸, ⊆) for some set 퐸. In particular, the Cauchy ideal completion CId(푃 ) of any partial order 푃 is a Priestley space. The morphisms between two Priestley spaces are the order-preserving and continuous maps or Priestley maps. The category of Priestley spaces with Priestley maps is denoted by 퐏퐫. A bounded distributive lattice, henceforth simply a lattice, is a partial order 퐿 which admits a maximal element ⊤, a minimal element ⊥, a greatest lower bound (infimum) 푎 ∧ 푏 and a least upper bound (supremum) 푎 ∨ 푏 for every 푎, 푏 ∈ 퐿 and satisfies the distributivity law 푎 ∧ (푏 ∨ 푐) = (푎 ∧ 푏) ∨ (푎 ∧ 푐), for every 푎, 푏, 푐 ∈ 퐿. Importantly, bounded distributive lattices are shown to form a variety of algebras in the signature (∧, ∨, ⊥, ⊤). A bounded lattice homomorphism, henceforth just a lattice homomorphism, is a map ℎ ∶ 퐿 → 퐾 preserving ⊤, ⊥, ∧ and ∨. We denote by 퐃퐋퐚퐭 the category of (bounded distributive) lattices with (bounded) lattice homomorphisms. A prime filter of a lattice 퐿 is a subset 푈 of 퐿 such that ⊥ ∉ 푈, ⊤ ∈ 푈, 푎, 푏 ∈ 푈 implies 푎 ∧ 푏 ∈ 푈, and for every 푎, 푏 ∈ 퐿, 푎 ∨ 푏 ∈ 푈 implies 푎 ∈ 푈 or 푏 ∈ 푈. For a lattice 퐿 we denote by 헗(퐿) the set of prime filters of 퐿 partially ordered by reverse inclusion and endowed with the topology Stone topology, namely the topology generated by the sets of the form {푈 ∈ 헗(퐿) ∣ 푎 ∈ 푈} and {푈 ∈ 헗(퐿) ∣ 푎 ∉ 푈}, for 푎 ∈ 퐿. Conversely for every Priesltey space 풳 we denote by 험(풳) the lattice of clopen downsets of 풳. Moreover, to each lattice homomorphism ℎ ∶ 퐿 → 퐾 corresponds an order- preserving and continuous map 헗(ℎ) ∶ 헗(퐾) → 헗(퐿) given by 헗(ℎ)(푈) = ℎ−1(푈) for every prime filter 푈 of the lattice 퐾. Conversely to each order- preserving continuous map 푓 ∶ 풳 → 풴 between Priestley spaces corresponds a lattice homomorphism 험(푓) ∶ 험(풴) → 험(풳) given by 험(푓)(퐷) = 푓−1(퐷) for every clopen downset 퐷 of 푌 . Theorem 4.17 (Priestley duality). The functors 헗 ∶ 퐃퐋퐚퐭 → 퐏퐫 and 험 ∶ 퐏퐫 → 퐃퐋퐚퐭 establish a duality of categories. In particular, 헗(험(풳)) and

82 4 The ideal space of a well-quasi-order

풳 are isomorphic as Priestley spaces and 험(헗(퐿)) and 퐿 are isomorphic as lattices. Remark 4.18. We could have chosen to equip the set of prime filters of a lattice with the order of inclusion (instead of reverse inclusion) and have defined the dual of a Priestley space to be the lattice of clopen upsets (instead of clopen downsets) with the same result. While this convention makes some of the statements look nicer, we will regret in Theorem 4.20, for example, not to have chosen the other convention. This duality encompasses two other famous dualities. Firstly, every finite partial order admits a unique topology, namely the discrete topology, which makes it a Priestley space. Priestley duality restricts to a duality between the category of finite partial orders with order-preserving maps and finite bounded distributive lattices: this is the content of Birkhoff duality [Bir37]. Secondly, Boolean algebras are particular examples of bounded distributive lattices. Since the prime filters of a Boolean algebra are the maximal filters, or ultrafilters, the partial order on the dual of a Boolean algebra under Priestley duality is discrete. The Priestley dual of a Boolean algebra, also called the Stone dual, is therefore simply a zero-dimensional compact Hausdorff space, or Boolean space. The restriction of Priestley duality to Boolean algebras on one side and Boolean spaces on the other is the celebrated Stone duality. Importantly, we recall that Priestley duality turns sub-objects into quotients and vice versa: Theorem 4.19. For any lattice homomorphism ℎ ∶ 퐿 → 퐾 the following are equivalent. (i) ℎ is injective if and only if 헗(ℎ) is surjective; (ii) ℎ is surjective if and only if 헗(ℎ) is both an order embedding and a topological embedding. Towards the identification of the Priestley dual of the Cauchy ideal completion of a partial order 푃 , we study a slightly more general situation where Priestley duality relates the taking of the topological closure with the algebraic generation of a lattice. Let 퐴 and 퐵 be sets and 푅 ⊆ 퐴 × 퐵 a binary relation from 퐴 to 퐵. Let ← 푅 (푏) = {푎 ∈ 퐴 ∣ 푎 푅 푏} for every 푏 ∈ 푇 . We denote by 푋푅 the Priestley space consisting in the closure of the set {푅←(푏) ∣ 푏 ∈ 퐵} inside 2퐴 partially ordered by inclusion. → Let 푅 (푎) = {푏 ∈ 퐵 ∣ 푎 푅 푏} for 푎 ∈ 퐴. We denote by 퐿푅 the bounded distributive lattice generated by {푅→(푎) ∣ 푎 ∈ 퐴} inside (풫(퐵), ∩, ∪, ∅, 퐵),

83 4 The ideal space of a well-quasi-order namely the smallest family of subsets of 퐵 which contains {푅→(푎) ∣ 푎 ∈ 퐴}, contains 퐵 and ∅, and is closed under both finite intersections and finite unions. op op We denote by 퐿푅 the opposite of the lattice 퐿푅. Observe that 퐿푅 is conveniently realised as the bounded distributive lattice generated by {¬푅→(푎) ∣ 푎 ∈ 퐴} inside (풫(퐵), ∩, ∪, ∅, 퐵), where ¬푅→(푎) = {푏 ∈ 퐵 ∣ ¬푎 푅 푏} = 퐵 ∖푅→(푎).

Theorem 4.20 ([BPZ07]). The lattice of clopen upsets of 푋푅 is isomorphic to the lattice 퐿푅, or dually 푋푅, is isomorphic to the space of prime filters of 퐿푅 ordered by inclusion and equipped with the Stone topology. Hence, with our op convention, 푋푅 is the Priestley dual of the lattice 퐿푅 . Bekkali, Pouzet, and Zhani [BPZ07] provide a direct proof of this fact. We now provide a ‘fast proof’ (cf. Figure 4.2) that we find to be conceptually more illuminating. It relies on the following fact. On the one hand, the space (2퐴, ⊆) is dual to the free bounded distributive lattice on the set 퐴 of free generators. On the other hand, the ‘free Priestley space’ associated to the set 퐵, namely the Stone–Czech compactification 훽퐵, is dual to the Boolean lattice 풫(퐵). This allows us to relate via Priestley duality the ‘taking of the topological closure’ and the process of ‘generating a lattice’.

훽푅←

퐴 2 푋푅 훽퐵

푅←

퐴 푅 퐵

푅→

퐹퐃퐋퐚퐭퐴 퐿푅 풫(퐵)

→ 퐹퐃퐋퐚퐭푅

Figure 4.2: Hamburger diagram: left vs right, algebra vs topology.

Proof. Consider the map 푅← ∶ 퐵 ⟶ 2퐴, 푏 ⟼ 푅←(푏) from the set 퐵 into the Priestley space 2퐴 partially ordered by inclusion. Now restricting the preimage map of 푅← to the lattice 퐹 of clopen upsets of 2퐴 we obtain the lattice

84 4 The ideal space of a well-quasi-order homomorphism (푅←)−1 ∶ 퐹 → 풫(퐵). The dual map of (푅←)−1 is the unique continuous map 훽푅← ∶ 훽퐵 → 2퐴 which extends 푅← to the Stone–Czech compactification of the discrete space 퐵. The image of 훽푅← inside 2퐴 is simply 푋푅. The lattice of clopen upsets of (2퐴, ⊆) is (isomorphic to) the free bounded distributive lattice 퐹퐃퐋퐚퐭(퐴) on the set 퐴 of free generators and to every 푎 ∈ 퐴 corresponds the clopen upset {푋 ∈ 2퐴 ∣ 푎 ∈ 푋}. The lattice homomorphism (푅←)−1 is therefore uniquely determined by its value on the sets {푋 ∈ 2퐴 ∣ 푎 ∈ 푋}, for 푎 ∈ 퐴, and we have

(푅←)−1({푋 ∈ 2퐴 ∣ 푎 ∈ 푋}) = {푏 ∈ 퐵 ∣ 푎 ∈ 푅←(푏)} = {푏 ∈ 퐵 ∣ 푎 푅 푏} = 푅→(푎).

It follows that (푅←)−1 is equal to the unique lattice homomorphism

→ 퐹퐃퐋퐚퐭(푅 ) ∶ 퐹퐃퐋퐚퐭(퐴) → 풫(퐵)

→ → which extends the map 푅 ∶ 퐴 → 풫(퐵). The image of 퐹퐃퐋퐚퐭(푅 ) in 풫(퐵) is of course the lattice 퐿푅. → Summarising, the lattice homomorphism 퐹퐃퐋퐚퐭푅 ∶ 퐹퐃퐋퐚퐭(퐴) ⟶ 풫(퐵) is dual to the order-preserving continuous map 훽푅← ∶ 훽퐵 ⟶ 2퐴. Whence the lattice of clopen upsets of 푋푅 is isomorphic to 퐿푅. We now apply Theorem 4.20 to the particular case of a reflexive, transitive and antisymmetric relation ⩽ ⊆ 푃 × 푃 . Notice that the Priestley space 푋⩽ is simply the Cauchy ideal completion CId(푃 ). The lattice 퐿⩽ generated by the {↑ 푝 ∣ 푝 ∈ 푃 } inside (풫(푃 ), 푃 , ∅, ∩, ∪) is called the tail lattice [BPZ07] of 푃 and is denoted by TailLat(푃 ).

Theorem 4.21. Let 푃 be a partial order. Then the Cauchy ideal completion CId(푃 ) is the Priestley dual of the opposite of TailLat(푃 ).

In fact, the property 푀 really pertains to the dual lattice of CId(푃 ).

Lemma 4.22. The following are equivalent for a partial order 푃 .

(i) 푃 has property 푀;

(ii) TailLat(푃 ) consists precisely in the finitely generated upsets of 푃 ;

op (iii) 풟fb(푃 ) = TailLat(푃 ) .

85 4 The ideal space of a well-quasi-order

Proof. Firstly notice that for any po 푃 the finitely generated upsets are closed under finite unions and contains the empty set – since ∅ = ↑ ∅ is finitely generated. (i)↔ (ii): If 푃 has property 푀 and 퐹 , 퐺 ⊆ 푄 are finite then it follows that the set ↑ 퐹 ∩ ↑ 퐺 = ⋃ {푝, 푞}↑ is finitely generated. Moreover 푃 = ∅↑ 푝∈퐹,푞∈퐺 is finitely generated if 푃 has property 푀. The converse follows from the fact that by definition 퐹 ↑ = ⋂ ↑ 푝. 푝∈퐹

(ii)↔ (iii): Notice that the family 풟fb(푄) is obtained by complementation from the family of finitely generated upsets. Whence for the partial orders which enjoy the property 푀 we have the following. Proposition 4.23. Let 푃 be a partial order with property 푀. Then the Cauchy ideal completion CId(푃 ), which coincides with Id(푃 ) equipped with the Lawson topology, is the Priestley dual of 풟fb(푃 ). We give the above mentioned isomorphism explicitly. Proposition 4.24. Let 푃 be a partial order with property 푀. The map

헗(풟fb(푃 )) ⟶ Id(푃 )

푈 ⟼ 퐼푈 = ⋂ 푈 = {푝 ∈ 푃 |푃 ∖ ↑ 푝 ∉ 푈} is an isomorphism between the Priestley dual of 풟fb(푃 ) and Id(푃 ) = CId(푃 ) whose inverse is given by

Id(푃 ) ⟶ 헗(풟fb(푃 ))

퐼 ⟼ 푈퐼 = {퐷 ∈ 풟fb(푃 ) ∣ 퐼 ⊆ 퐷}.

Proof. Let 푈 be a prime filter of 풟fb(푃 ). Let us first show that ⋂ 푈 = {푝 ∈ 푃 ∣ 푃 ∖ ↑ 푝 ∉ 푈}. Clearly if 푃 ∖ ↑ 푝 ∈ 푈, then 푝 ∉ ⋂ 푈. Conversely, if 푝 ∉ ⋂ 푈, then there exists 퐷 ∈ 푈 with 푝 ∉ 퐷. Therefore 퐷 ⊆ 푃 ∖ ↑ 푝, and since 푈 is a filter 푃 ∖ ↑ 푝 belongs to 푈 too. Notice that if 푃 is the empty partial order, then both the dual of 풟fb(푃 ) = {∅} and Id(푃 ) are empty. So suppose that 푃 is not empty and that 푈 is a prime filter of 풟fb(푃 ). Clearly 퐼푈 is a downset of 푃 . We show that 퐼푈 is an ideal of 푃 . Since 푃 has property 푀 we have 푃 = ↑ 퐹 for some non-empty finite 퐹 ⊆ 푃 . Since ∅ = ⋂ 푃 ∖ ↑ 푟 does not belong to the prime filter 푈 푟∈퐹 there exists 푟 ∈ 퐹 with 푃 ∖ ↑ 푟 ∉ 푈, and so 푟 ∈ ⋂ 푈 and 퐼푈 is non empty.

86 4 The ideal space of a well-quasi-order

Moreover 퐼푈 is directed as follows from Proposition 4.4 and the fact that 푈 is prime. The map 푈 ↦ 퐼푈 is clearly an order embedding. To see it is continuous, observe that for every finite 퐹 , 퐺 ⊆ 푃 and every 푈 ∈ 헗(풟fb(푃 )):

퐼푈 ∈ 푁(퐹 , 퐺) ⟷ ∀푝 ∈ 퐹 푃 ∖ ↑ 푝 ∉ 푈 and ∀푞 ∈ 퐺 푃 ∖ ↑ 푞 ∈ 푈. As a continuous and injective map from a compact space to a Hausdorff space, 푈 ↦ 퐼푈 is therefore a homeomorphism onto its image. To see it is onto Id(푃 ), let 퐼 be an ideal of 푃 . Let us show that 푈퐼 is a prime filter of 풟fb(푃 ). Clearly 푈퐼 is upward closed and closed under finite intersections. Since 퐼 is non empty, ∅ ∉ 푈퐼. Finally the fact that 푈퐼 is prime follows from the directedness of 퐼 via Proposition 4.4 again. Returning to the case of a wqo, we recall that a partial order 푃 is wqo if and only if every upset of 푃 is finitely generated. Whence in this case, the lattice 풟fb(푃 ) of finitely bounded downsets is simply equal to the lattice 풟(푃 ) of downsets of 푃 . Therefore when 푃 is wqo, the Cauchy ideal completion is the Priestley dual of the lattice 풟(푃 ). As a matter of fact, for an arbitrary partial order 푃 the Priestley dual of the lattice 풟(푃 ) of downsets turns out to be the profinite completion of 푃 , also known as the Nachbin order-compactification of 푃 .

4.1.4 Profinite completion, Nachbin order-compactification We now describe another very natural completion of a partial order called the profinite completion. This completion is very natural from a category theory perspective. Though the material of this subsection is certainly considered folklore by experts, we were unable to find any reference. We therefore provide proofs for these basic results. Let us denote by 퐏퐨 the category of partial orders together with order- preserving maps. The profinite completion of a partial order 푃 is in a sense ‘the best approximation of 푃 among finite partial orders’. As it is, such abest approximation does not exist among finite partial orders, the reason being arguably that the category 퐏퐨fin of finite partial orders is only finitely complete, but not complete. To remedy this and make sense of the above intuitive idea, one consider the pro-completion Pro- 퐏퐨fin of the category 퐏퐨fin, which is somehow the completion of 퐏퐨fin with respect to the property of admitting all (small) limits. The category Pro- 퐏퐨fin admits as objects directed diagrams 퐷 ∶ 퐼 → 퐏퐨fin and the morphisms between 퐷 ∶ 퐼 → 퐏퐨fin and 퐸 ∶ 퐽 → 퐏퐨fin are given by

87 4 The ideal space of a well-quasi-order the following double limit formula

lim lim{푓 ∶ 퐷(푖) → 퐸(푗) ∣ 푓 is order-preserving}. 퐽 퐼

Fortunately the category Pro- 퐏퐨fin is equivalent to the concrete category 퐏퐫 of Priestley spaces [Joh86, (VI 3.3), p. 248; Pri94; Spe72]. In this equivalence a directed diagram 퐷 ∶ 퐼 → 퐏퐨fin corresponds to the actual limit lim퐼 퐷(푖) taken in the category of ordered topological spaces, regarding each finite partial order 퐷(푖) as a discrete space, and the morphisms between profinite partial orders are realised as the order-preserving and continuous maps, the Priestley maps. The profinite completion of a partial order 푃 is – now formulated correctly – the best approximation (from the left) of 푃 among the profinite partial orders, or equivalently, among the Priestley spaces. In formal terms, the profinite completion of partial orders 헣 ∶ 퐏퐨 → 퐏퐫 is the left adjoint to the forgetful functor 헨 ∶ 퐏퐫 → 퐏퐨 which simply drops the topology. We now describe concretely this functor. For a partial order 푃 we denote by ℰ푃 the set of all quasi-orders 퐸 on 푃 which contains ⩽ and whose partial order quotient is finite, i.e. the reflexive and transitive relations 퐸 ⊆ 푃 × 푃 such that

(1) for all 푝, 푞 ∈ 푃 , 푝 ⩽ 푞 implies 푝 퐸 푞,

(2) the quotient 푃 /퐸 by the equivalence relation (푝 퐸 푞 and 푞 퐸 푝) has finitely many classes.

Now when partially ordered by inclusion ℰ푃 is directed, since 퐸, 퐹 ∈ ℰ푃 implies that 퐸 ∩ 퐹 ∈ ℰ. Moreover each time 퐸 ⊆ 퐹 for 퐸, 퐹 ∈ ℰ푃 there is a unique order-preserving and surjective map 휋퐸,퐹 ∶ 푃 /퐸 → 푃 /퐹 , [푝]퐸 → [푝]퐹 . We let

퐷푃 ∶ ℰ푃 ⟶ 퐏퐨fin 퐸 ⟼ 푃 /퐸

퐸 ⊆ 퐹 ⟼ 휋퐸,퐹 ∶ 푃 /퐸 → 푃 /퐹 be the directed diagram of finite quotients of 푃 .

Proposition 4.25. Let 푃 be a partial order and 푃̂ be the limit in 퐏퐫 of the ̂ diagram 퐷푃 , regarding finite partial orders as finite Priestley spaces. Then 푃 is the Priestley dual of the lattice 풟(푃 ).

88 4 The ideal space of a well-quasi-order

′ Proof. By Birkhoff duality, consider the diagram 퐷푃 ∶ ℰ푃 → 퐃퐋퐚퐭fin dual ′ −1 to 퐷푃 , namely 퐷푃 (퐸) = 풟(푃 /퐸) for every 퐸 ∈ ℰ푃 and 퐸 ⊆ 퐹 ↦ 푝퐸,퐹 ∶ ′ 풟(푃 /퐹 ) → 풟(푃 /퐸). Let 퐷 be the colimit of 퐷푃 in 퐃퐋퐚퐭. This distributive lattice is just the lattice 풟(푄) of downsets of 푄. Indeed 퐷 is really the union of all downsets of 푄 of the form 푝−1(퐴) for an order-preserving map 푝 ∶ 푄 → 푅 with 푅 finite and 퐴 a downset of 퐹 . But for every downset 퐿 of 푄 the function 푝퐿 ∶ 푄 → 2 with 푝퐿(푞) = 0 ↔ 푞 ∈ 퐿 into 2 = {0 < 1} is order-preserving and −1 ′ 휒퐿 ({0}) = 퐿. By Priestley duality the colimit of 퐷푃 in 퐃퐋퐚퐭 is dual to the ̂ limit of 퐷푃 . Therefore 푃 is the Priestley dual of 풟(푃 ). From this description of 푃̂ we get the following embedding

푒 ∶ 푃 ⟶ 푃̂ 푝 ⟼ ̂푝= {퐷 ∈ 풟(푃 ) ∣ 푝 ∈ 퐷}.

We generally identify 푃 with 푒(푃 ) = { ̂푝∣ 푝 ∈ 푃 }.

Lemma 4.26. Let 푃 be a partial order, then 푃̂ is the closure of { ̂푝∣ 푝 ∈ 푃 } inside 2풟(푃 ) and so 푃 is dense in 푃̂. Proof (folklore). Suppose that 푈 ∈ 2풟(푃 ) belongs to the closure of { ̂푝∣ 푝 ∈ 푃 }. Since {푋 ∈ 2풟(푃 ) ∣ ∅ ∈ 푋} is open and ∅ ∉ ̂푝 for all 푝 ∈ 푃 , ∅ ∉ 푈. If 퐷, 퐸 ∈ 푈 and 퐷 ∩ 퐸 ∉ 푈, then 푈 belongs to the basic open set 푁({퐷, 퐸}, {퐷 ∩ 퐸}) of 2풟(푃 ), and so there exists 푝 ∈ 푃 with ̂푞∈ 푁({퐷, 퐸}, {퐷 ∩ 퐸}), i.e. 푝 ∈ 퐷, 푝 ∈ 퐸, and 푝 ∉ 퐷 ∩ 퐸, contradiction. Hence 퐷, 퐸 ∈ 푈 implies 퐷 ∩ 퐸 ∈ 푈. Next if 퐷 ∈ 푈 and 퐸 ∉ 푈, then 푈 ∈ 푁({퐷}, {퐸}) and thus there exists 푝 ∈ 푃 with 푝 ∈ 퐷 and 푝 ∉ 퐸, so 퐷 ⊈ 퐸. Finally if 퐷 ∪ 퐸 ∈ 푈 and 퐷, 퐸 ∉ 푈, then 푈 ∈ 푁({퐷 ∪ 퐸}, {퐷, 퐸}) so there is 푝 ∈ 푃 with 푝 ∈ 퐷 ∪ 퐸 and 푝 ∉ 퐷, 푝 ∉ 퐸, a contradiction. So every 푈 ∈ 2풟(푃 ) which belongs to the closure of 푒(푃 ) is a prime filter on 풟(푃 ). Conversely let 푈 be a prime filter on 풟(푃 ) and 퐷0, … , 퐷푛, 퐸0, … 퐸푚 ∈ 풟(푃 ) be such that 푈 ∈ 푁({퐷0, … , 퐷푛}, {퐸0, … , 퐸푚}). Then since 푈 is a prime filter, we have 퐷 = ⋂푛 퐷 ∈ 푈 and 퐸 = ⋃푚 퐸 ∉ 푈, so necessarily 푖=0 푖 푗=0 푗 퐷 ⊈ 퐸. Hence there exists 푝 ∈ 퐷 ∖ 퐸 and we have

̂푝∈ 푁({퐷0, … , 퐷푛}, {퐸0, … , 퐸푚}). Therefore 푈 belongs to the closure of 푒(푃 ). Proposition 4.27. Let 푃 be a partial order. For every Priestley space 푋 and every order-preserving map 푓 ∶ 푃 → 푋 there exists a unique Priestley map 푓̂ ∶ 푃̂ → 푋 such that 푔 ∘ 푒 = 푓.̂

89 4 The ideal space of a well-quasi-order

헨(푋) 푋 푓

푃 헨(푓)̂ 푓 ̂ 푒 헨(푃̂ ) 푃̂

Proof. Let 험(푋) be the lattice of clopen downsets of 푋. Since 푓 ∶ 푃 → 푋 is order-preserving the preimage map goes 푓−1 ∶ 풟(푋) → 풟(푃 ) and restricts to a homomorphism 푓−1 ∶ 험(푋) → 풟(푃 ) whose Priesltey dual is an order- preserving continuous map 푓̂ ∶ 푃̂ → 푋. Clearly we have 푓̂ ∘ 푒 = 푓. Finally if ̂ ̂ ℎ ∶ 푃 → 푋 is a Priestley map such that ℎ ∘ 푒 = 푓, then ℎ↾푃 = 푓 = 푓↾푃 . Since both ℎ and 푓 ̂ are continuous and agree the subset 푃 of 푃̂ which is dense by Lemma 4.26, we must have 푓̂ = ℎ. This allows us to define the functor

헣 ∶ 퐏퐨 ⟶ 퐏퐫 푃 ⟼ 푃̂ (푓 ∶ 푃 → 푄) ⟼ (푓̂ ∶ 푃̂ → 푄)̂ where 푓 ̂ is the dual map of 푓−1 ∶ 풟(푄) → 풟(푃 ) under Priestley duality. By the previous proposition, this functor is left adjoint to the forgetful functor 헨 ∶ 퐏퐫 → 퐏퐨 as desired. Notice that for a discrete partial order (푋, =) the profinite completion 푋̂ is just the Stone-Čech compactification of the set 푋. Indeed 풟(푋) = 풫(푋) and the Priestley dual of the complete Boolean algebra 풫(푋) is simply its Stone dual. In particular, the profinite completion 퐴̂ of the antichain 퐴 = (휔, =) is the Stone-Čech compactification 훽휔 of the natural numbers. Actually, the profinite completion of a partial order enjoys a stronger universal property as notably proved by Bezhanishvili et al. [Bez+06].

Proposition 4.28 (Bezhanishvili et al. [Bez+06, Proposition 3.4]). Let 푃 be a partial order. For every compact Hausdorff space 푌 equipped with a partial order ⩽ closed in 푌 × 푌 and every order preserving map 푓 ∶ 푃 → 푌 there exists a unique continuous and order-preserving map 푓̂ ∶ 푃̂ → 푌 such that 푓̂ ∘ 푒 = 푓.

90 4 The ideal space of a well-quasi-order

푓 푌

푃 푓 ̂ 푒 푃̂

This proposition shows that the profinite completion of a partial order 푃 is also the Nachbin order-compactification [Nac65, p.103–104] of 푃 equipped with the discrete topology. We also notice that the profinite completion functor behaves nicely with respect to embeddings.

Lemma 4.29. Let 푓 ∶ 푃 → 푄 be an order-preserving map between posets. Then 푓 is an order embedding if and only if 푓̂ ∶ 푃̂ → 푄̂ is both an order embedding and a topological embedding.

Proof. By Theorem 4.19 it is enough to prove that 푓 ∶ 푃 → 푄 is an order embedding if and only if 푓−1 ∶ 풟(푄) → 풟(푃 ) is surjective. Notice that if 푓 is −1 an embedding, then for every downset 퐷 ∈ 풟(푃 ) we have 퐷 = 푓 ↓푄 푓(퐷) since

′ −1 ′ 푝 ∈ 푓 ↓푄 푓(퐷) ⟷ ∃푝 ∈ 퐷 푓(푝 ) ⩽푄 푓(푝) ′ ⟷ ∃푝 ∈ 퐷 푝 ⩽푃 푝 ⟷ 푝′ ∈ 퐷.

Conversely suppose 푓−1 ∶ 풟(푄) → 풟(푃 ) is surjective and let 푝, 푝′ ∈ 푃 with 푓(푝) ⩽ 푓(푝′). There exists 퐷 ∈ 풟(푄) with 푓−1(퐷) = ↓ 푝′ and since 푓(푝) ⩽ 푓(푝′) ∈ 퐷 we have 푝 ∈ ↓ 푝′, i.e. 푝 ⩽ 푝′. Back to wqos, we have obtained:

Corollary 4.30. Let 푃 be a partial order. If 푃 is wqo, then the following are isomorphic as partially ordered topological spaces:

(i) the ideal completion Id(푃 ) with the Lawson topology,

(ii) the Cauchy ideal completion CId 푃 ,

(iii) the Priestley dual of 풟(푃 ),

(iv) the profinite completion 푃̂,

91 4 The ideal space of a well-quasi-order

(v) the Nachbin order-compactification of 푃 .

If one wants to view the profinite completion 푃̂ of a wqo 푃 as Id(푃 ), then the action of the functor 헣 ∶ 퐏퐨 → 퐏퐫 on order-preserving maps between wqos can be simply described as follows:

Lemma 4.31. Let 푓 ∶ 푃 → 푄 be an order preserving map between wqos. The Priestley map 헣(푓) ∶ 푃̂ → 푄̂ viewed as a map 헣(푓) ∶ Id(푃 ) → Id(푄) is given by

헣(푓)(퐼) = ↓푄{푓(푝) ∣ 푝 ∈ 퐼} for every 퐼 ∈ Id(푃 ). Proof. The Priestley map 헣(푓) ∶ Id(푃 ) → Id(푄) is the dual of the lattice homomorphism 푓−1 ∶ 풟(푄) → 풟(푃 ). Now an ideal 퐼 ∈ Id(푃 ) corresponds to the prime filter 푈퐼 = {퐷 ∈ 풟(푃 ) ∣ 퐼 ⊆ 퐷} via the isomorphism given in −1 Proposition 4.24. Then 푉퐼 = {퐸 ∈ 풟(푄) ∣ 푓 (퐸) ∈ 푈퐼} is the prime filter of ̂ ̂ 풟(푄) which is the image of 푈퐼 by 헣(퐹 ) ∶ 푃 → 푄. Now, by Proposition 4.24 again, the prime filter 푉퐼 corresponds to the ideal of 푄 given by {푞 ∈ 푄 ∣ 푄 ∖ ↑ 푞 ∉ 푉퐼}. We therefore have

헣(푓)(퐼) ={푞 ∈ 푄 ∣ 푄 ∖ ↑ 푞 ∉ 푉퐼} ={푞 ∈ 푄 ∣ 퐼 ⊈ 푓−1(푄 ∖ ↑ 푞)} ={푞 ∈ 푄 ∣ ∃푝 ∈ 퐼 푓(푝) ∉ 푄 ∖ ↑ 푞} ={푞 ∈ 푄 ∣ ∃푝 ∈ 퐼 푓(푝) ⩾ 푞}

= ↓푄{푓(푝) ∣ 푝 ∈ 퐼}, as desired. Remark 4.32. The canonical extension of a bounded distributive lattice is concretely defined as the lattice of downsets of its Priestley dual. However, italso admits an algebraic characterisation as the unique complete distributive lattice in which the lattice embeds in a dense and compact way (see for example the nicely written survey by Gehrke and Vosmaer [GV11]). For a wqo 푄, the wqo character of Id(푄) is naturally related to the canonical extension of 풟(푄). Namely if 푄 is wqo, then Id(푄) is wqo if and only if the canonical extension of 풟(푄) is well-founded. Indeed Id(푄) is the Priestley dual of 풟(푄), so the lattice of downsets of Id(푄) is canonical extension of 풟(푄). Moreover Id(푄) is wqo if and only if 풟(Id(푄)) is well-founded by Proposition 2.6.

92 4 The ideal space of a well-quasi-order

4.2 From well to better: A proof of Pouzet’s conjecture

Henceforth 푄 stands for a wqo and we assume 푄 is a partial order. The space of ideals of 푄 is the partially ordered compact space described by one of the equivalent alternatives gathered in Corollary 4.30. The partial order 푄 is viewed as a subset of Id(푄) via the embedding 푞 ↦ ↓ 푞. The space Id(푄), being a compact Hausdorff topological space, admits a unique uniformity which agrees with its topology. As a subset of Id(푄) the wqo 푄 is equipped with the induced uniformity. We henceforth refer to this uniformity on 푄 when we talk about a Cauchy, or uniformly continuous, super-sequence 푓 ∶ 퐹 → 푄 into a wqo. Remark 4.33. It follows from Lemma 4.9 and Proposition 3.13 that a super- sequence 푓 ∶ 퐹 → 푄 into a wqo 푄 is Cauchy if and only if for every 푞 ∈ 푄 we have 푓−1(↑ 푞) ∈ Blocks(퐹 ). Theorem 4.34 (with R. Carroy). Let 푄 be wqo. Then every super-sequence 푓 ∶ 퐹 → 푄 admits a Cauchy sub-super-sequence 푓′ ∶ 퐹 ′ → 푄 which therefore extends to a continuous map 푓′ ∶ 퐹 ′ → Id(푄). Proof. As 푄 is wqo, the ideal space Id(푄) is a compact subspace of 2푄. Now suppose first that 푄 is countable, then 2푄 is metrisable and therefore so is Id(푄). In this case, the statement follows from Theorem 3.25. In the general case, even though 푄 may not be countable, the set 푃 = Im 푓 always is. By Lemma 4.29, the embedding of 푃 into 푄 extends to an embedding of Id(푃 ) into Id(푄) both in the order and the topological sense. Therefore 푓 ∶ 퐹 → 푄 is uniformly continuous if and only if 푓 ∶ 퐹 → 푃 is uniformly continuous. Since 푃 is wqo as a subset of a wqo, the first case applies. Remark 4.35. In the previous proof, the only place where the hypothesis that 푄 is wqo, and not only has property 푀, is used is in the general case. This is when we conclude that 푃 is wqo (or in fact that it has property 푀 for that matter) from the fact that 푄 is wqo. This is indeed necessary since as we already noticed the property 푀 is not hereditary. As a corollary we have new characterisation of bqo: Corollary 4.36. A quasi-order 푄 is bqo if and only if 푄 is wqo and every Cauchy super-sequence is good. The cofinality of an ideal 퐼 ∈ Id(푄), denoted cof(퐼), is the least cardinal 휆 such that there exists a subset 퐵 of 퐼 of cardinality 휆 with 퐼 = ↓ 퐵. The

93 4 The ideal space of a well-quasi-order principal ideals are the ideals of cofinality 1, they are the only ideals with finite cofinality. We denote respectively by Id⩽ω(푄) and Idω(푄) the po of ideals of 푄 which have countable cofinality and the po of ideals with cofinality 휔. Observe * that Id⩽ω(푄) = 푄 ∪ Idω(푄) and that Idω(푄) ⊆ Id (푄). The following should not come as a surprise. Lemma 4.37. If 푓 ∶ 퐹 → 푄 is a Cauchy super-sequence into a wqo, then the the image of the unique continuous extension 푓̄ ∶ 퐹 → Id(푄) is contained in Id⩽ω(푄).

Proof. Since 퐹 is metrisable, for every 푠 ∈ 퐹 there exists a sequence (푠푛)푛∈휔 ⊆ 퐹 which converges to 푠. Then by continuity of 푓 we have 푓(푠) = lim 푓(푠푛). Then by Proposition 4.16, there is 푁 ∈ [휔]∞ such that 푓(푠) = ⋃ ↓ 푓(푠 ). 푛∈푁 푛 Therefore 푓(푠)̄ has countable cofinality. Before proving the main result of this chapter we stop on our crucial example once again. Example 4.38 (Rado’s poset). Rado’s partial order ℜ was defined in Ex- ample 2.15, it is given by the set [휔]2 partially ordered by :

푚 = 푚′ and 푛 ⩽ 푛′, or {푚, 푛} ⩽ {푚′, 푛′} ⟷ { 푛 < 푚′.

We claim that Id(ℛ) = ℛ ∪ {퐼푛 ∣ 푛 ∈ 휔} ∪ {⊤} where for all 푛 ∈ 휔 퐼푛 = ⋃ ↓(푛, 푘) and ⊤ = ℛ. We have (푚, 푛) ⩽ 퐼 if and only if 푚 = 푘 or 푛 < 푘, 푛<푘 푘 and 푎 ⩽ ⊤ for all 푎 ∈ Id(ℛ). The non principal ideals are the 퐼푛s and ⊤. We show there are no other ideals. Let 퐼 be an ideal of ℛ. First suppose for all 푘 ∈ 휔 there is an (푚, 푛) ∈ 퐼 with 푘 < 푚, then 퐼 = ⊤. Suppose now that there exists 푚 = max{푘 ∣ ∃푙(푘, 푙) ∈ 퐼}. If there is infinitely many 푛 such that (푚, 푛) ∈ 퐼 then 퐼 = 퐼푚. Otherwise 퐼 = ↓(푚, 푛) for 푛 = max{푙 ∣ (푚, 푙) ∈ 퐼}. In Example 2.15 we showed that ℜ is wqo but not bqo as notably witnessed by the bad super-sequence id ∶ [휔]2 → ℛ, which is the identity on the underlying sets. Using Remark 4.33, it is clear that id is Cauchy and therefore extends to a continuous map id ∶ [휔]⩽2 → Id(ℛ). This continuous extension is simply given by id({푚}) = 퐼푚 for every 푚 ∈ 휔 and id(∅) = ⊤ (see Figure 4.3). Now the restriction of id to the barrier [휔]1 is a bad sequence in Id*(ℛ) witnessing the fact that it is not wqo. Hence this Cauchy bad super-sequence into ℛ yields a bad super-sequence into the non principal ideals of ℛ. Theorem 4.39. Let 푓 ∶ 퐹 → 푄 be a Cauchy super-sequence into a wqo and 푓̄ ∶ 퐹 → Id(푄) its continuous extension. If 푓 is bad, then there exists a front

94 4 The ideal space of a well-quasi-order

•

• • • • • • • ⋯ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋱ • • • • • • • • • • • • • id • • • • • • • • • • • • • • 2 The barrier [휔] • The po Id(ℛ)

Figure 4.3: A bad Cauchy super-sequence into Rado’s poset.

̄ * ̄ 퐺 ⊆ 퐹 such that the restriction 푓↾퐺 ∶ 퐺 → Id (푄) is bad. Moreover 푓↾퐺 has image in Idω(푄). Proof. By going to 푓̌ ∶ 퐹̌ → 푄 if necessary by Proposition 2.53, we can assume that 푓 ∶ 퐹 → 푄 is spare. By Theorem 3.31 we can assume that 푓 ∶ 퐹 → Id(푄) is −1 * normal. Since Isol(Id(푄)) = 푄 by Proposition 4.15 we have Λ푓 = 푓 (Id (푄)). Moreover by Lemma 3.36 we obtain 푓−1(Id*(푄)) = 퐹 ∖ 퐹 . If 푓−1(Id*(푄)) is empty, then 퐹 is trivial and thus 푓 is good, a contradiction. Otherwise 푓−1(Id*(푄)) is the closure of a front 퐺 on 푋 = ⋃ 퐹 and by Lemma 3.37 we have 퐹 = {푠 ∪ {푛} ∣ 푠 ∈ 퐺 and 푛 ∈ 푋/푠}. If 퐺 is trivial then 퐹 = [푋]1, since 푄 is wqo, then 푓 is good, a contradiction again. ̄ * Otherwise 퐺 is not trivial, and consider 푔 = 푓↾퐺 ∶ 퐺 → Id (푄). By contradiction, suppose there is no front 퐺′ ⊆ 퐺 such that the restriction ′ * ′ 푔↾퐺′ ∶ 퐺 → Id (푄) is bad. Hence by Corollary 2.63 there exists a front 퐺 ⊆ 퐺 ′ ′ such that the restriction 푔↾퐺′ ∶ 퐺 → 푄 is perfect. Let 푠 ∈ 퐺 be minimal for ⊆ ′ ′ in 퐺 and 푌 = ⋃ 퐺 . By continuity we have 푔(푠) = lim푛∈푌 /푠 푓(푠∪{푛}) and by Proposition 4.16 there is 푍 ∈ [푌 /푠]∞ such that 푔(푠) = {푞 ∈ 푄 ∣ ∃푛 ∈ 푍 푞 ⩽ ′ 푓(푠 ∪ {푛})}. For 푛0 = min 푍 there exists 푡 ∈ 퐺 with 푡 ⊏ ∗푠 ∪ {푛0} ∪ 푌 /푛0. ′ By the minimality of 푠 in 퐺 for the inclusion, we have ∗푠 ∪ {푛0} ⊑ 푡. Again ′ ∞ 푔(푡) = lim푛∈푌 /푡 푓(푡 ∪ {푛}) and by Proposition 4.16 there is 푍 ∈ [푌 /푡] with 푔(푡) ={푞 ∈ 푄 ∣ ∃푚 ∈ 푍′ 푞 ⩽ 푓(푠 ∪ {푚})}. Since 푠 ¡ 푡 in 퐺′ and

95 4 The ideal space of a well-quasi-order

′ 푔 is perfect, we have 푔(푠) ⊆ 푔(푡). Therefore there exists 푚0 ∈ 푍 with ¡ 푓(푠 ∪ {푛0}) ⩽ 푓(푡 ∪ {푚0}). Since 푠 ∪ {푛0} 푡 ∪ {푚0}, 푓 is good. This contradiction terminates the proof. As a direct corollary we have the proof of the conjecture by Pouzet [Pou78].

Theorem 4.40 (with R. Carroy). Let 푄 be wqo. If Idω(푄) is bqo, then 푄 is bqo. In particular if Id*(푄) is bqo, then 푄 is bqo. The following classes of wqos are sometimes considered as approximations of the concept of bqo.

Definition 4.41. Let 푄 be a quasi-order and 1 ⩽ 훼 < 휔1. We say that 푄 is 훼-bqo if and only if every super-sequence 푓 ∶ 퐹 → 푄 with rk 퐹 ⩽ 훼 is good. Clearly a qo is wqo if and only if it is 1-bqo, and it is bqo if and only if its 훼-bqo for every 훼 < 휔1. Remark 4.42. Marcone [Mar94], Pouzet and Sauer [PS06], for example, use a different definition of 훼-bqo which is easily seen to be equivalent to ours. A finer utilisation of Theorem 4.39 yields the following: Theorem 4.43. Let 푄 be a quasi-order.

(i) For every 훼 < 휔1, if 푄 is (훼 + 1)-bqo, then 풟(푄) is 훼-bqo and so Id*(푄) is 훼-bqo too.

(ii) If 푄 is wqo, then for every 훼 < 휔1, Idω(푄) is 훼-bqo implies that 푄 is (1 + 훼)-bqo.

(iii) If 푄 is wqo then for every 푛 < 휔, 푄 is (1+푛)-bqo if and only if Idω(푄) is 푛-bqo. Proof. (i) By contraposition, suppose 푓 ∶ 퐹 → 풟(푄) is a bad super-sequence from some front 퐹 . A bad super-sequence 푓2 ∶ 퐹 2 → 푄 from the front 퐹 2 = {푠 ∪ 푡 ∣ 푠, 푡 ∈ 퐹 and 푠 ¡ 푡} is obtained by choosing for every 푠, 푡 ∈ 퐹 with 푠 ¡ 푡 some 푓(푠 ∪ 푡) ∈ 2 2 푓(푠)∖푓(푡). By observing that for each ray 퐹푛, the tree 푇 (퐹푛) is included 2 in the union 푇 (퐹 ) ∪ 푇 (퐹푛), we see that rk 퐹 ⩽ rk 퐹 + 1. (ii) By contraposition, suppose 푄 is wqo and there is bad super-sequence from a front 퐹 into 푄, then by Theorem 4.39 there exists a bad super- sequence from some front 퐺 with 퐺 ⊆ 퐹 ∖퐹 into Idω(푄). By Lemma 3.34 we have 1 + rk 퐺 ⩽ rk 퐹 , so if rk 퐹 ⩽ 1 + 훼 then rk 퐺 ⩽ 훼.

96 4 The ideal space of a well-quasi-order

(iii) Since for a finite ordinal 푛 we have 푛 + 1 = 1 + 푛, the statement follows from (i) and (ii). Remark 4.44. In an attempt to prove the converse of Theorem 4.43 (ii), one could try the following. For a bad super-sequence 푓 ∶ 퐹 → Idω(푄), one could consider a map 푓+ ∶ 퐹 + → 푄 defined on the front 퐹 + = {푠 ∪ {푛} ∣ 푠 ∈ 퐹 and 푛 ∈ 휔/푠}

푠 푠 by choosing a strictly increasing sequence (푞푛)푛∈휔/푠 with ↓{푞푛 ∣ 푛 ∈ 휔/푠} = 푠 푓(푠) and letting 푓(푠 ∪ {푛}) = 푞푛 for every 푠 ∈ 퐹 and 푛 ∈ 휔/푠. Notice that rk 퐹 + = 1 + rk 퐹 as desired. However assuming that 푓+ is perfect is not sufficient to prove that 푓 is good, as the following example shows. Consider the super-sequence 푓+ ∶ [휔]2 → ℜ into Rado’s partial order given by 푓+(푚, 푛) = (2푚, 푚 + 푛). The super-sequence 푓+ is perfect since whenever 푚 < 푛 < 푘 we have (2푚, 푚 + 푛) ⩽ (2푛, 푛 + 푘) in ℜ. Using Remark 4.33, 푓+ is easily seen to be Cauchy and therefore extends to a continuous 푓̄ ∶ [휔]⩽2 → ̄ * Id(ℜ). But clearly the restriction 푓 = 푓↾[휔]1 → Id (ℜ) is bad. This observation is a motivation to seek to strengthen the notion of ‘perfect super-sequence’ in the conclusion of the implication ‘if there is no bad sequence in 푄, then every super-sequence admits a perfect sub-super-sequence’ while maintaining the validity of this fact. The results presented in Section 2.3 can be seen as a first attempt in this direction. Although some ingredients seem yet to be missing, we make the following conjecture. Notice that it differs from the claim made by Pouzet and Sauer [PS06, Theorem 2.17].

Conjecture 2. Let 푄 be wqo and 훼 < 휔1. Then 푄 is (1 + 훼)-bqo if and only if Idω(푄) is 훼-bqo.

4.3 Corollaries

4.3.1 Finitely many non principal ideals The first corollary of Theorem 4.40 that we mention is: Corollary 4.45. If 푄 is wqo and Id*(푄) is finite, then 푄 is bqo. This result is due to Pouzet [Pou78] and a direct proof is presented by Fraı̈ssé [Fra00, Chapter 7, 7.7.8]. This first simple corollary already allows us to prove the following proposition, a particular case of which was used by Carroy [Car13].

97 4 The ideal space of a well-quasi-order

Proposition 4.46. Let 휑 ∶ 휔 → 휔 be progressive, i.e. such that 푛 ⩽ 휑(푛) for every 푛 ∈ 휔. Then the partial order ⩽휑 on 휔 defined by

푚 ⩽휑 푛 ⟷ 푚 = 푛 or 휑(푚) < 푛. is a better-quasi-order. Proof. Let 푔 ∶ 휔 → 휔 be any sequence. Then either 푔 is bounded in the usual order and so 푔 is good for ⩽휑, or 푔 is unbounded in the usual order and so there exists 푛 such that 푔(푛) > 휑(푔(0)) and so 푔 is good for ⩽휑. Hence (휔, ⩽휑) is wqo. Now let 퐼 be a non principal ideal in (휔, ⩽휑). In particular 퐼 is an infinite subset of 휔, so for every 푚 ∈ 휔 there exists 푛 ∈ 퐼 such that 휑(푚) < 푛 and so 푚 ⩽휑 푛 ∈ 퐼. Therefore 퐼 = 휔 and so there is exactly one non-principal ideal of (휔, ⩽휑). It follows by Corollary 4.45 that (휔, ⩽휑) is bqo.

4.3.2 Interval orders We now turn to the case of a wqo 푄 such that Id*(푄) is a well order. Since by Proposition 2.54 well orders are bqo, such quasi-orders are bqo by The- orem 4.40. Observe that when 푄 is wqo, since ideals are downsets and 풟(푄) is well- founded, Id*(푄) is well-founded too. Henceforth, if 푄 is wqo then Id*(푄) is linearly ordered if and only if Id*(푄) is a well order. What are the quasi-orders whose non principal ideals are linearly ordered? Well, assume 푄 is a quasi-order and that 퐼, 퐽 ∈ Id*(푄) are incomparable for inclusion. Let 푝 ∈ 퐼 ∖ 퐽 and 푞 ∈ 퐽 ∖ 퐼. Then 푝 is incomparable with 푞. Forbidding antichains of size 2 in 푄 is simply asking that 푄 is a linear order, and of course well orders are bqo. But we can do better: since 퐼 and 퐽 are non principal, there are 푝′ ∈ 퐼 with 푝 < 푝′ and 푞′ ∈ 퐽 with 푞 < 푞′. The restriction of the quasi-order on 푄 to {푝, 푞, 푝′, 푞′} is isomorphic to the po • • 2 ⊕ 2 = • • and therefore 2 ⊕ 2 embeds into 푄. We are naturally led to following definition which appears frequently in the literature. Definition 4.47. A partial order 푃 is an interval order if the partial order 2 ⊕ 2 does not embed into 푃 . In other words for every 푝, 푞, 푥, 푦 ∈ 푃 , 푝 < 푥 and 푞 < 푦 imply 푝 < 푦 or 푞 < 푥.

98 4 The ideal space of a well-quasi-order

The preceding discussion yields the following which is already stated by Pouzet and Sauer [PS06]. Theorem 4.48. An interval order is bqo if and only if it is wqo. Notice that this theorem can be rephrased as follows: a partial order 푃 such that neither (휔, =), nor 휔op (the opposite of 휔), nor 2 ⊕ 2 embeds into 푃 is a better-quasi-order. According to Fishburn and Monjardet [FM92], the notion of interval order was first studied by the twenty-years-old Norbert Wiener [Wie14] who credits Bertrand Russell for suggesting the subject. Wiener was later acknowledged as the originator of cybernetics [CS06]. The reverse mathematics of interval orders is studied by Marcone [Mar07]. For 푝 ∈ 푃 , let Pred(푝) = {푞 ∈ 푃 ∣ 푞 < 푝}. It is easy to see that a partial order 푃 is an interval order if and only if the set {Pred(푝) ∣ 푝 ∈ 푃 } is linearly ordered by inclusion. The terminology ‘interval order’ was introduced by Fishburn [Fis70] and stems from the following characterisation. A non trivial closed interval of a partial order 푄 is a set of the form [푎, 푏] = {푞 ∈ 푄 ∣ 푎 ⩽ 푞 ⩽ 푏} for some 푎, 푏 ∈ 푄 with 푎 < 푏. We partially order the set Int(푄) of non trivial closed intervals of 푄 by [푎, 푏] ⩽ [푐, 푑] if and only if 푎 = 푐 and 푏 = 푑 or 푏 ⩽ 푐. For a partial order 푃 let us say that a map 퐼 ∶ 푃 → Int(푄) is an interval representation of 푃 in 푄 if for every 푥, 푦 ∈ 푃 we have 푥 < 푦 ↔ 퐼(푥) < 퐼(푦). Let us first see that any partial order 푃 admits an interval representation. Let Pred+(푝) = ⋂ Pred(푥) and 푝<푥 + 푄푃 = {Pred(푝) ∣ 푝 ∈ 푃 } ∪ {Pred (푝) ∣ 푝 ∈ 푃 } be partially ordered by inclusion. Proposition 4.49 ([Bog93]). Let 푃 be a partial order. The map

퐼 ∶ 푃 ⟶ Int(푄푃 ) + 푝 ⟼ 퐼푝 = ( Pred(푝), Pred (푝)) is an interval representation of 푃 in 푄푃 . Proof. First observe that for every 푝 ∈ 푃 we have Pred(푞) ⊂ Pred+(푞) since 푞 < 푝 imply 푞 ∈ Pred(푥) for all 푥 > 푝, and in fact 푝 ∈ Pred+(푝) ∖ Pred(푝). So 퐼 is well defined. If 푝 < 푞, then Pred+(푝) = ⋂ Pred(푥) ⊆ Pred(푞), 푝<푥 + and so 퐼푝 < 퐼푞. Conversely if 퐼푝 < 퐼푞, then Pred (푝) ⊆ Pred(푞) and since 푝 ∈ Pred+(푝) we have 푝 < 푞. Hence 퐼 is an interval representation of 푃 .

99 4 The ideal space of a well-quasi-order

The following is a slight generalisation of a theorem by Fishburn [Fis70]. The proof we give is due to Bogart [Bog93].

Proposition 4.50. A partial order 푃 is an interval order if and only if there exists an interval representation of 푃 in some linear order.

Proof. Suppose 퐼 ∶ 푃 → Int(퐿) is an interval representation of 푃 in a linear order 퐿 and let 푝0 < 푝1 and 푞0 < 푞1 in 푃 . If 퐼(푝푖) = [푙푖, 푟푖] and 퐼(푞푖) = [푚푖, 푠푖] then 푟0 ⩽ 푙1 and 푠0 ⩽ 푚1. Since 퐿 is linearly ordered, either 푟0 ⩽ 푚1 and so 푝0 < 푞1, or 푚1 ⩽ 푟0 and so 푞0 < 푝1. Therefore 푃 is an interval order. Conversely, suppose 푃 is an interval order. By Proposition 4.49, it suffices to prove the 푄푃 is linearly ordered. But {Pred(푝) ∣ 푝 ∈ 푃 } is linearly ordered and Pred+(푝) = ⋂ Pred(푥) is incomparable for the inclusion with some 푝<푥 푋 ∈ 푄푃 if and only if Pred(푥) is incomparable with 푋 for some 푥 > 푝.

4.3.3 Classes of better-quasi-orders via forbidden patterns In fact continuing the above discussion we find that for any qo 푄, Id*(푄) is linearly ordered if and only if the po ⋮ ⋮

휔 ⊕ 휔 = • • • • • • does not embed into 푄. We therefore have the following:

Theorem 4.51. If neither (휔, =), nor 휔op, nor 휔 ⊕ 휔 embed into 푄, then 푄 is bqo.

Suppose now for a partial order 푃 that there exists a natural number 푛 such that the size of every antichain of 푃 is bounded by 푛. Then, by a theorem due to Dilworth [Dil50], for 퐴 an antichain of maximum size, say 푛, there exist subsets 푃 , 푖 ∈ 푛, such that |푃 ∩퐴| = 1, 푃 is linearly ordered and ⋃ 푃 = 푃 푖 푖 푖 푖∈푛 푖 (see also [Fra00, 4.14.1, p. 141]). In particular, if 푃 is further assumed to be well-founded, then 푃 is bqo as a finite union of well orders. Continuing further the discussion of the previous subsection, we see that if there exists an antichain 퐴 of size 푛 among the non principal ideals of a qo 푄, then the partial order

100 4 The ideal space of a well-quasi-order

⋮ ⋮ ⋮

• • • 푛 ⊗ 휔 = • • ⋯ • • • • ⎭ } } } } ⎬ } } } } ⎫ 푛 times embeds into 푄. Indeed, assume that {퐼푖 ∣ 푖 ∈ 푛}, 푛 ⩾ 2 is an antichain of non principal ideals of a qo 푄. For each 푖 ∈ 푛 and every 푗 ∈ 푛 with 푖 ≠ 푗, since 퐼푖 ⊈ 퐼푗 we can pick 푞푗 ∈ 퐼푖 ∖ 퐼푗 and by the fact that 퐼푖 is directed there is 푖 푖 푞 ∈ 퐼푖 with 푞 ∉ 퐼푗 for every 푗 ≠ 푖. Now since each 퐼푖 is non principal there 푖 푖 푖 exists a strictly increasing sequence (푞푘)푘∈휔 in 퐼푖 with 푞0 = 푞 . This clearly yields and embedding of 푛 ⊗ 휔 into ⋃ 퐼 . Therefore 푖∈푛 푖 Theorem 4.52. Let 푛 ⩾ 1. If neither (휔, =), nor 휔op, nor 푛 ⊗ 휔 embed into 푄, then 푄 is bqo.

In this theorem, for each 푛 ⩾ 1, we have a class of bqo which is defined by finitely many forbidden patterns. Examples of classes of bqos defined by mean of forbidden patterns – left alone by finitely many – are quite rare. In fact to our knowledge the previous theorem is the best result of this sort.

101 5 A Wadge hierarchy for second countable spaces

Our last chapter is of a more applied nature. We are interested in quasi-ordering the subsets of a topological space according to their complexities. Among the properties of such a quasi-order the following are arguably wished for. It should agree with an a priori idea of topological complexity, in particular it should refine the classical hierarchies of topological complexity. Moreover it shouldbe as fine as possible while still being wqo or even bqo – at least on the Borel subsets. The Wadge quasi-order on the Baire space 휔휔 satisfies all these properties and even more. But the fact that the Wadge quasi-order is well-founded on Borel subsets of 휔휔 is not at all straightforward. It relies on the determinacy of certain infinite games and this result is actually best seen as an immediate corollary of a theorem on bqos obtained by van Engelen, Miller, and Steel [vEMS87, Theorem 3.2]. Elaborating on Chapter 2, we start this chapter by presenting a slight generalisation of this theorem in a way which makes it appear as an extension of the very idea underlying the definition of bqo. We then define a notion of reducibility based on relatively continuous relations. We show that this notion of reducibility satisfies the above mentioned properties and generalises the Wadge quasi-order to virtually all second countable 푇0 spaces. This chapter is based on an article [Peq15] published by the author in Archive for Mathematical Logic.

5.1 The van Engelen–Miller–Steel theorem

In Chapter 2, we started from the idea that a qo 푄 is bqo if the quasi-order 푉 ∗(푄) of non empty sets over 푄 is wqo. We then found a convenient definition of bqo by observing that any bad sequence in 푉 ∗(푄) yields a multi-sequence into 푄 of a particular kind, which we identified as the bad multi-sequences. Pushing the same idea a little further we showed that when 푄 is bqo, then so is 푉 ∗(푄). We then observed in Subsection 2.2.5 that the restriction to

102 5 A Wadge hierarchy for second countable spaces the quasi-order 퐻∗ (푄) of hereditarily countable non-empty subsets over 푄 휔1 already contained all the required information. In particular, we showed in Theorems 2.58 and 2.59 that 푄 is bqo if and only if 퐻∗ (푄) is wqo if and 휔1 only if 퐻∗ (푄) is well-founded. We now describe 퐻∗ (푄) in slightly different 휔1 휔1 terms before we greatly generalise this construction in a way that is crucial to the present chapter. We can endow a set 퐴 with the discrete topology in which every subset of 퐴 is open. This space 퐴 admits the compatible metric 훿(푎, 푏) = 1 whenever 푎 ≠ 푏. The set 퐴휔 of infinite sequences on 퐴, viewed as the product space of infinitely many copies of 퐴, is also metrisable: we use the compatible ultramet- −푛 ric 푑(푥, 푦) = 2 if 푥 ≠ 푦 and 푛 is the least number with 푥푛 ≠ 푦푛. When 퐴 is countable, 퐴휔 is a Polish space, i.e. a separable completely metrisable space. Important examples are given by the Cantor space 2휔 and the Baire space 휔휔. Recall from Definitions 2.28 that by a tree on a non-empty set 퐴 we mean a set 푇 ⊆ 퐴<휔 of finite sequences on 퐴 which is closed under prefixes, i.e. 푢 ⊑ 푣 and 푣 ∈ 푇 imply 푢 ∈ 푇 . We say that a tree 푇 on 퐴 is pruned if for every 푢 ∈ 푇 there exists 푣 ∈ 푇 with 푢 ⊏ 푣. For a tree 푇 on 퐴, we let

휔 [푇 ] = {푥 ∈ 퐴 ∣ for every 푛 ∈ 휔, 푥↾푛 ∈ 푇 } denote the set of infinite branches of 푇 . It is well-known that the map 푇 ↦ [푇 ] is a one-to-one correspondence between pruned trees on 퐴 and closed subsets of 퐴휔. ∗ For 푋 ∈ 푉 (푄) recall that tcQ 푋 is the smallest transitive set containing {푋} when elements of 푄 are treated as urelements or atoms, namely each element of 푄 contains no element and is different from the empty set. Notice that in particular for 푞 ∈ 푄 we have tcQ(푞) = {푞}. Therefore by the axiom of countable choice we have 푋 ∈ 퐻∗ (푄) if and only if 푋 ∈ 푉 ∗(푄) and tc (푋) 휔1 Q is countable. For every 푋 ∈ 퐻∗ (푄) we define a non-empty pruned tree 푇 on tc (푋) by 휔1 푋 Q letting

<휔 (1) if 푋 ∈ 푄, then 푇푋 = {푋} ,

(2) if 푋 ∉ 푄, then 푇푋 is the the set of finite sequences 푠 on tcQ(푋) such that

푠 = ∅ or [푠0 ∈ 푋 and ∀푖 < |푠| − 1

(푠푖 ∋ 푠푖+1 or (푠푖 ∈ 푄 and 푠푖 = 푠푖+1))].

103 5 A Wadge hierarchy for second countable spaces

Next we define a map 푙푋 ∶ [푇푋] → 푄 by letting for every 훼 ∈ [푇푋]

푙푋(훼) = 푞 ⟷ ∃푛 ∈ 휔 훼푛 = 푞.

By the axiom of foundation 푙푋 is defined on the whole of [푇푋]. Moreover 푙푋 −1 is locally constant, so 푙푋 (푞) is open in [푇푋] for every 푞 ∈ 푄 and 푙푋 has a countable image. For every 푋 ∈ 퐻∗ (푄) the map 푙 ∶ [푙 ] → 푄 is – up to a bijection between 휔1 푋 푋 휔 tcQ(푋) and 휔 – a locally constant map from a closed subset of 휔 into 푄. We now generalise 퐻∗ (푄) in the following way. Recall that the family of 휔1 Borel subsets of a topological space 풳 is the smallest Boolean algebra of subsets of 풳 containing the open sets and closed under countable union and complementation – and hence under countable intersection. Recall that if 풴 is a subspace of 풳 then the family of Borel sets of 풴 consists of the intersection with 풴 of some Borel set of 풳. Definition 5.1. Let 푄 be a quasi-order. A map 푙 ∶ 퐷 → 푄 from a Borel subset 퐷 of 휔휔 is called a Borel 푄-labelling function on 퐷 if (1) for every 푞 ∈ 푄, the set 푙−1(푞) is Borel in 퐷,

(2) the image Im 푙 of 푙 is countable. We refer to 퐷 as the domain of 푙 and denote it by dom 푙. We let 픏퐵(푄) be the set of all Borel 푄-labelling functions on some Borel subset of 휔휔. 퐵 For every 푙0 ∶ 퐷0 → 푄 and 푙1 ∶ 퐷1 → 푄 in 픏 (푄) we define a two player game with perfect information 퐺(푙0, 푙1) as follows (see Figure 5.1). The players I and II choose natural numbers alternatively. Player I starts by choosing some 훼0 ∈ 휔, then Player II chooses some 훽0 ∈ 휔, then it is again the turn of Player I to choose some 훼1 ∈ 휔, so on and so forth. Both players eventually produce 훼 ∈ 휔휔 and 훽 ∈ 휔휔 respectively. Player II wins the play (훼, 훽) if and only if the following condition holds:

훼 ∈ 퐷0 implies (훽 ∈ 퐷1 and 푙0(훼) ⩽ 푙1(훽)). Player I wins the play (훼, 훽) if and only if Player II does not win, namely when the following condition holds: ⩽̸ 훼 ∈ 퐷0 and (훽 ∈ 퐷1 implies 푙0(훼) 푙1(훽)). We then quasi-order 픏퐵(푄) by

푙0 ⩽픏 푙1 ⟷ Player II has a winning strategy in 퐺(푙0, 푙1).

104 5 A Wadge hierarchy for second countable spaces

퐺(푙 , 푙 ) ⎩ { { ⎨ 0 { 1{ ⎧ I II

훼0 훽0

훼1 훽1

훼2 훽2

⋮ ⋮

훼 훽

Figure 5.1: A play of the game 퐺(푙0, 푙1).

퐵 Remark 5.2. When 푙0 and 푙1 in 픏 (푄) have non-empty closed domains, then there are non-empty pruned trees 푇0 and 푇1 on 휔 with dom 푙0 = [푇0] and dom 푙1 = [푇1]. The game 퐺(푙0, 푙1) is then equivalent to the following game where each player is required to stay inside 푇0 or 푇1 respectively. Player I starts by choosing some 훼0 ∈ 휔 with (훼0) ∈ 푇0, then Player II chooses some 훽0 ∈ 휔 with (훽0) ∈ 푇1, then Player I chooses some 훼1 ∈ 휔 such that (훼0, 훼1) ∈ 푇0, so on and so forth. Both players eventually produce 훼 ∈ [푇0] and 훽 ∈ [푇1] respectively. Player II wins if and only if 푙0(훼) ⩽ 푙1(훽) in 푄. Clearly the restriction of any winning strategy for II in 퐺(푙0, 푙1) to the positions for I which belong to 푇0 is a winning strategy for Player II in the above game. Conversely the restriction of any winning strategy for Player I in 퐺(푙0, 푙1) to the positions for II which belong to 푇1 is a winning strategy for I in the above game. In particular it is clear from the remark that for every 푋, 푌 ∈ 퐻∗ (푄) the 휔1 game 퐺푉 ∗ (푋, 푌 ) (see Subsection 2.2.4) is equivalent to the game 퐺(푙푋, 푙푌 ). Therefore 푋 ↦ 푙 is an embedding of 퐻∗ (푄) into 픏퐵(푄) and we can really 푋 휔1 see 픏퐵(푄) as a Borel generalisation of 퐻∗ (푄). 휔1 In the definition of 픏퐵(푄) the restriction to Borel 푄-labelling functions on 휔 Borel subsets of 휔 ensures that the game 퐺(푙0, 푙1) is Borel and therefore determined by Martin’s Borel determinacy [Mar75]. By the axiom of choice,

105 5 A Wadge hierarchy for second countable spaces such a restriction is actually necessary to get positive results. However one can weaken this restriction if one is willing to assume the determinacy of a larger class of games. As observed for the first time by van Engelen, Miller, and Steel[vEMS87, Theorem 3.2] a bad locally constant multi-sequence in 픏퐵(푄) can be reflected into 푄, in a similar way as we reflected a bad locally constant multi-sequence in 푉 ∗(푄) or 퐻∗ (푄) into 푄 in Proposition 2.55. But this time we do not get 휔1 a locally constant multi-sequence into 푄 in general, but only a Borel multi- sequence. Definition 5.3. A multi-sequence 푓 ∶ [푋]∞ → 퐸 into some set 퐸 is said to be Borel if 푓 has a countable image and 푓−1(푒) is Borel in [푋]∞ for every 푒 ∈ 퐸. The following can be seen as the Borel analogue of Proposition 2.55 and its proof follows the same line (see the discussion in Subsection 2.2.4). This is a slight generalisation of a theorem by van Engelen, Miller, and Steel [vEMS87, Theorem 3.2] (see also the paper by Louveau and Saint Raymond [LS90, The- orem 3]). Theorem 5.4. Let 푄 be a qo. For every bad locally constant multi-sequence ℎ ∶ [휔]∞ → 픏퐵(푄) there exists a bad Borel multi-sequence 푔 ∶ [휔]∞ → 푄 such that 푔(푋) ∈ Im(ℎ(푋)) for every 푋 ∈ [휔]∞.

∞ 퐵 Proof. Let ℎ ∶ [휔] → 픏 (푄), 푋 ↦ ℎ푋 be a bad locally constant multi- sequence. Let 퐿 be the countable image of ℎ inside 픏퐵(푄) and endow 퐿 with the discrete topology. We consider the closed subset of 퐿휔 of ‘nowhere ascend- ing’ sequences in 퐿: na 휔 ⩽̸ 퐿 = {(푙푛)푛∈휔 ∈ 퐿 ∣ for every 푛 ∈ 휔, 푙푛 Lip 푙푛+1}.

na ̃ na na ̃ Notice that 퐿 comes with a shift map 햲 ∶ 퐿 → 퐿 given by 햲((푙푛)푛∈휔) = ∞ (푙푛+1)푛∈휔. We are going to define 푔 ∶ [휔] → 푄 as the composition of a map ℎ⃗ ∶ [휔]∞ → 퐿na with a map Ψ ∶ 퐿na → 푄. ∞ ∞ 0 Recall that 햲 ∶ [휔] → [휔] denotes the shift map 푋 ↦ ∗푋 and let 햲 = 푘+1 푘 id[휔]∞ and 햲 = 햲 ∘ 햲 for every 푘 ∈ 휔. ⃗ ∞ na ⃗ 푛 Claim. The map ℎ ∶ [휔] → 퐿 defined by ℎ(푋) = ℎ(햲 (푋))푛∈휔 is continuous and we have ℎ⃗ ∘ 햲 = 햲̃ ∘ ℎ⃗. 푛 ⩽̸ 푛+1 ⃗ Since ℎ is bad, ℎ(햲 (푋)) Lip ℎ(햲 (푋)) for every 푛 ∈ 휔 and so ℎ is well 휔 th defined. If we let 휋푛 ∶ 푃 → 푃 denote the projection on the 푛 coordinate, ⃗ 푛 then the map 휋푛 ∘ ℎ = ℎ(햲 (푋)) is continuous since 햲 is continuous and ℎ is locally constant. It follows that ℎ⃗ is continuous and we have proved the Claim.

106 5 A Wadge hierarchy for second countable spaces

Claim. There is a map Ψ ∶ 퐿na → 푄 such that

(i) for every 휆 ∈ 퐿na we have Ψ(휆) ⩽̸ Ψ(햲(휆))̃ in 푄,

na (ii) Ψ(휆) ∈ Im 푙0 for every 휆 = (푙푛)푛∈휔 ∈ 퐿 , (iii) the image of Ψ is countable and Ψ−1(푞) is Borel in 퐿na for every 푞 ∈ 푄. ⩽̸ 퐵 If 푙0 픏 푙1 in 픏 (푄), then II has no winning strategy in the Borel game 퐺(푙0, 푙1) and therefore Player I has a winning strategy. By the axiom of countable choice we choose for every 푙 , 푙 ∈ 퐿 with 푙 ⩽̸ 푙 a winning strategy 휎푙1 0 1 0 픏 1 푙0 for Player I in 퐺(푙0, 푙1).

퐺(푙 , 푙 ) 퐺(푙 , 푙 ) 퐺(푙 , 푙 ) ⎩ { { { ⎨ 0 { 1{ { ⎧ ⎩ { { { ⎨ 2 { 3{ { ⎧ ⎩ { { { ⎨ 4 { 5{ { ⎧

퐺(푙 , 푙 ) 퐺(푙 , 푙 ) ⎩ { { { ⎨ 1 { 2{ { ⎧ ⎩ { { { ⎨ 3 { 4{ { ⎧

푙 푙 푙 푙 푙 푙0 휎 1 푙1 휎 2 푙2 휎 3 푙3 휎 4 푙4 휎 5 푙5 푙0 푙1 푙2 푙3 푙4

0 푙 1 푙 2 푙 3 푙 4 훼0 휎 1 훼0 휎 2 훼0 휎 3 훼0 휎 4 훼0 푙0 푙1 푙2 푙3

0 푙 1 푙 2 푙 3 훼1 휎 1 훼1 휎 2 훼1 휎 3 훼1 ⋮ 푙0 푙1 푙2

0 푙 1 푙 2 훼2 휎 1 훼2 휎 2 훼2 ⋮ 푙0 푙1

0 푙 1 훼3 휎 1 훼3 ⋮ 푙0

0 훼4 ⋮

⋮

0 1 2 3 4 훼 ∈ dom 푙 훼 ∈ dom 푙 훼 ∈ dom 푙 훼 ∈ dom 푙 훼 ∈ dom 푙

0 1 2 3 4

Figure 5.2: Stringing strategies together towards infinity.

107 5 A Wadge hierarchy for second countable spaces

na 푖 For each sequence 휆 = (푙푛)푛∈휔 ∈ 퐿 we define a diagram (훼 (휆))푖∈휔 ∈ ∏ dom 푙 by stringing strategies together as depicted in Figure 5.2. In 푖∈휔 푖 each game 퐺(푙 , 푙 ) Player I follows the strategy 휎푙푛+1 and Player II cop- 푛 푛+1 푙푛 푛 푛 ies the moves made by Player I in 퐺(푙푛+1, 푙푛+2). Writing 훼 (휆) = (훼푘 )푘∈휔 푛 this means that for every 푛, 푘 ∈ 휔 the number 훼푘 is the next move for Player I in 퐺(푙 , 푙 ) provided by the strategy 휎푙푛+1 in the position where 푛 푛+1 푙푛 푛+1 푛 푛+1 Player II has played 훼 ↾푘. Therefore, for every 푛 ∈ 휔, (훼 (휆), 훼 (휆)) is a play of 퐺(푙 , 푙 ) in which Player I has followed the winning strategy 휎푙푛+1 푛 푛+1 푙푛 푛 and so necessarily 훼 (휆) ∈ dom 푙푛. It follows that for every 푛 ∈ 휔 we have 푛 ⩽̸ 푛+1 푙푛(훼 (휆)) 푙푛+1(훼 (휆)). na 0 na We define Ψ ∶ 퐿 → 푄 by Ψ(휆) = 푙0(훼 (휆)) for every 휆 = (푙푛)푛∈휔 ∈ 퐿 . na We have Ψ((푙푛)푛∈휔) ∈ Im 푙0 for every (푙푛)푛∈휔 ∈ 퐿 and therefore the image of Ψ is contained in the countable union of countable sets ⋃ Im 푙. For 푙 ∈ 퐿, 푙∈퐿 na na na let us denote by 퐿푙 = {(푙푛)푛 ∈ 퐿 ∣ 푙0 = 푙} the clopen subset of 퐿 of 0 na sequences starting with 푙. For every 푙 ∈ 퐿, the map 훼 ↾ na ∶ 퐿푙 → dom 푙 is 퐿푙 na 0 continuous since for every 휆 = (푙푛)푛 ∈ 퐿푙 the initial segment 훼 (휆)↾푘 depends only on 푙 , … , 푙 – and it is determined by 휎푙1 , 휎푙2 … 휎푙푘 – as easily seen by 1 푘 푙0 푙1 푙푘−1 contemplating Figure 5.2. Therefore for every 푞 ∈ 푄, −1 −1 0 Ψ (푞) = ⋃ (푙 ∘ 훼 ↾ na ) (푞) 퐿푙 푙∈퐿 0 is Borel since 퐿 is countable, 푙 is Borel and 훼 ↾ na is continuous. 퐿푙 na 1 Moreover, by construction for every 휆 = (푙푛)푛∈휔 ∈ 퐿 , we have 훼 (휆) = 훼0(햲(휆))̃ and therefore 0 ⩽̸ 1 0 ̃ ̃ Ψ(휆) = 푙0(훼 (휆)) 푙1(훼 (휆)) = 푙1(훼 (햲(휆))) = Ψ(햲(휆)), as 휎푙1 is winning for Player I in 퐺(푙 , 푙 ). This finishes the proof of the second 푙0 0 1 Claim. Finally we define the multi-sequence 푔 ∶ [휔]∞ → 푄 by letting 푔 = Ψ ∘ ℎ⃗. We have for every 푋 ∈ [휔]∞ 푔(푋) = Ψ(ℎ(푋))⃗ ⩽̸ Ψ(햲(̃ ℎ(푋)))⃗ = Ψ(ℎ(햲(푋))),⃗ hence 푔 is bad. Moreover 푔 is a Borel multi-sequence since for every 푞 ∈ 푄, the set 푔−1(푞) = ℎ⃗ −1(Ψ−1(푞)) is Borel in [휔]∞ since Ψ−1(푞) is Borel in 퐿na and ℎ⃗ preserves Borel sets by preimages since it is continuous.

108 5 A Wadge hierarchy for second countable spaces

The previous theorem only yields a Borel multi-sequence and we are therefore led to consider the combinatorial properties of Borel subsets of [휔]∞. Notice that Nash-Williams Theorem 2.36 is equivalent to the statement that for every clopen subset 퐶 of [휔]∞ there exists 푋 ∈ [휔]∞ such that either [푋]∞ ⊆ 퐶 or [푋]∞ ∩ 퐶 = ∅. Galvin and Prikry [GP73] proved the following important generalisation:

Theorem 5.5 (Galvin and Prikry). For every Borel subset 퐵 of [휔]∞ there exists 푋 ∈ [휔]∞ such that either [푋]∞ ⊆ 퐵 or [푋]∞ ∩ 퐵 = ∅.

The proof of this theorem uses an important technique sometimes called combinatorial forcing. Several textbooks (notably [Tod10; Kec95; Hal11; Jec03]) provide a proof of this theorem and its generalisation to the sets enjoying the Baire property relatively to the Ellentuck topology. In our context, we are interested in the following well-known corollary (see also [PV92, Theorem 3.4], [LS82]).

Theorem 5.6. Let 퐸 be a set. Every Borel multi-sequence into 퐸 admits a locally constant sub-multi-sequence. In particular a quasi-order 푄 is bqo if and only if there is no bad Borel multi-sequence into 푄.

∞ 푖 Proof (Folklore). Let 푔 ∶ [휔] → 푄 be a Borel multi-sequence and let (퐵 )푖∈휔 enumerate the countable Borel partition {푓−1(푞) ∣ 푞 ∈ Im 푔} of [휔]∞. We ∞ ∞ show that there exists 푋 ∈ [휔] such that for each 퐵푘 ∩ [푋] is clopen in [푋]∞, therefore showing that the sub-multi-sequence 푔 ∶ [푋]∞ → 푄 is locally ∞ constant. To do so, we define a sequence (푁푖)푖∈휔 ⊆ [휔] with 푛푖 = min 푁푖 such that for every 푖 ∈ 휔:

∞ (i) 푁푖+1 ∈ [∗푁푖] ,

(ii) for every 푠 ⊆ {푛푗 ∣ 푗 < 푖}

∞ 푖 ∞ 푖 either {푠 ∪ 푍 ∣ 푍 ∈ [푁푖] } ⊆ 퐵 , or {푠 ∪ 푍 ∣ 푍 ∈ [푁푖] } ∩ 퐵 = ∅,

∞ ∞ By Theorem 5.5, there exists 푁0 ∈ [휔] such that either [푁0] ⊆ 퐵0 or ∞ 퐵0 ∩ [푁0] = ∅. Now assume the sequence (푁푖)푖 is defined up to some 푖 ⩾ 0. ∞ For each 푠 ⊆ {푛푗 ∣ 푗 < 푖 + 1} we consider the following Borel subset of [∗푁푖] :

푖+1 ∞ 푖+1 퐵푠 = {푍 ∈ [∗푁푖] ∣ 푠 ∪ 푍 ∈ 퐵 }.

푖+1 Let 푠푙 for 푙 < 2 enumerate the subsets of {푛푗 ∣ 푗 < 푖 + 1}. Applying Theorem 5.5 to 퐵푖+1 we find 푌 ∈ [ 푁 ]∞ such that either [푌 ]∞ ⊆ 퐵푖+1 or 푠0 0 ∗ 푖 0 푠0

109 5 A Wadge hierarchy for second countable spaces

[푌 ]∞ ∩ 퐵푖+1 = ∅. Next we apply Theorem 5.5 to 퐵푖+1 to find 푌 ∈ [푌 ]∞ such 0 푠0 푠1 1 0 that either [푌 ]∞ ⊆ 퐵푖+1 or [푌 ]∞ ∩퐵푖+1 = ∅. We go on like this for 2푖+1 steps 0 푠1 0 푠1 and we let 푁푖+1 = 푌2푖+1 which satisfies the requirements. Finally let 푋 = {푛푖 ∣ 푖 ∈ 휔}. For any 푖 ∈ 휔 let 푆푖 denote the finite set of ∞ 푖 those 푠 ⊆ {푛푗 ∣ 푗 < 푖} such that {푠 ∪ 푍 ∣ 푍 ∈ [푁푖] } ⊆ 퐵 . Then for every 푖 ∈ 휔 we have 푖 ∞ ∞ 퐵 ∩ [푋] = ⋃ 푀푠 ∩ [푋] , 푠∈푆푖 ∞ where we recall that 푀푠 = {푌 ∈ [휔] ∣ 푠 ⊏ 푌 } is a basic clopen set. Therefore each 퐵푖 ∩ [푋]∞ is clopen in [푋]∞ as desired. Therefore we have obtained the following:

Theorem 5.7. Let 푄 be a quasi-order. If 푄 is bqo, then 픏퐵(푄) is bqo.

Remark that since 퐻∗ (푄) embeds into 픏퐵(푄) via 푋 ↦ 푙 , it follows from 휔1 푋 Theorem 2.59 that if 픏퐵(푄) is well-founded (in particular if it is wqo), then 푄 is bqo. Hence in a sense a quasi-order 푄 relates to 픏퐵(푄) in the same way as it does to 퐻∗ (푄). 휔1 In the game 퐺(푙0, 푙1), any winning strategy 휎 for Player II induces continuous ∗ 휔 휔 ∗ maps 휎 ∶ 휔 → 휔 such that for every 훼 ∈ dom 푙0 we have 휎 (훼) ∈ dom 푙1 and ∗ ∗ 푙0(훼) ⩽ 푙1(휎 (훼)). In particular the function 휎 restricts to a continuous map 푓 ∶ dom 푙0 → dom 푙1 such that for every 훼 ∈ dom 푙0 we have 푙0(훼) ⩽ 푙1(푓(훼)). This allows us to bring 픏퐵(푄) to the topological setting. A Luzin space – or Borel absolute space – is a separable metrisable space which is Borel in every Polish space in which it embeds. It is well-known a space 풳 is Luzin if and only if there exists a continuous bijection 푓 ∶ 퐹 → 풳 from a closed subset 퐹 of 휔휔, if and only if it is homeomorphic to a Borel subspace of some Polish space (see for example the article by Dellacherie [Del80, p.208]). Since every separable metrisable zero-dimensional space – or equivalently every zero-dimensional second countable space – is homeomorphic to a subspace of the Baire space, the zero-dimensional Luzin spaces are the topological spaces which are homeomorphic to a Borel subspace of 휔휔.

Definition 5.8. Let 푄 be a quasi-order. A map 푙 ∶ 풳 → 푄 from some Luzin zero-dimensional space 풳 is called a Borel 푄-labelling function of 풳 if

(1) for every 푞 ∈ 푄, the set 푙−1(푞) is Borel in 풳,

(2) the image Im 푙 of 푙 is countable.

110 5 A Wadge hierarchy for second countable spaces

We refer to 풳 as the domain of 푙 and denote it by dom 푙. We let 퐵(푄) be the class of all Borel 푄-labelling of some zero-dimensional Luzin space. We quasi-order 퐵(푄) by

푙0 ⩽c 푙1 ⟷ there exists a continuous map 푔 ∶ dom 푙0 → dom 푙1 s.t.

for every 푥 ∈ dom 푙0 we have 푙0(푥) ⩽ 푙1(푔(푥)) in 푄. Theorem 5.9. If 푄 is bqo, then 퐵(푄) is bqo. Proof. Let ℎ ∶ [휔]∞ → 퐵(푄) be a locally constant multi-sequence. Since ℎ has a countable image and every zero-dimensional Luzin space is homeomorphic to a Borel subset of 휔휔, we can assume that in fact ℎ ∶ [휔]∞ → 픏퐵(푄). By ∞ Theorem 5.7, if 푄 is bqo then there exist 푋 ∈ [휔] such that ℎ(푋) ⩽픏 ℎ(∗푋), i.e. Player II has a winning strategy in 퐺(ℎ(푋), ℎ(∗푋). Now a winning ∗ 휔 strategy 휎 for Player II in 퐺(ℎ(푋), ℎ(∗푋) induces a continuous map 휎 ∶ 휔 → 휔 휔 which restricts to a continuous 푓 ∶ dom ℎ(푋) → dom ℎ(∗푋) such that ℎ(푋)(훼) ⩽ ℎ(∗푋)(푓(훼)) for every 훼 ∈ dom ℎ(푋). Therefore ℎ(푋) ⩽c ℎ(∗푋) and ℎ ∶ [휔]∞ → 퐵(푄) is good. Since ℎ was arbitrary, it follows that 퐵(푄) is bqo. The case of the bqo 2 = {0, 1} partially ordered by equality has received much attention. In this case, 퐵(2) consists of the characteristic functions of Borel subsets of zero-dimensional Luzin spaces quasi-ordered by continuous reducibility, and by Theorem 5.9 퐵(2) is bqo. Let us write any 푙 ∈ 퐵(2) as −1 the ordered pair (퐴푙, dom 푙) where 퐴푙 = 푙 (1). Conversely we can write every Borel subset 퐴 of some zero-dimensional Luzin space 풳 as the element of 퐵(푄) consisting in its characteristic function 푙퐴 ∶ 풳 → 2. Now for every 푙0, 푙1 ∈ 퐵(푄) we have 푙0 ⩽c 푙1 if and only if there exists a continuous function 푓 ∶ dom 푙0 → dom 푙 such that 퐴 = 푓−1(퐴 ). We therefore also write (퐴, 풳) ⩽ (퐵, 풴) 1 푙0 푙1 W in place of 푙퐴 ⩽c 푙퐵. The symmetry of the qo 2 allows us to prove the following result, usually referred to as the ‘Wadge Lemma’. Lemma 5.10 (Wadge Lemma). Let 풳 and 풴 be zero-dimensional Luzin spaces, 퐴 ⊆ 풳 and 퐵 ⊆ 풴 be Borel. Then either (퐴, 풳) ⩽W (퐵, 풴) or (퐵, 풴) ⩽W (풳 ∖ 퐴, 풳).

휔 Proof. We can assume that 풳 and 풴 are Borel subsets 퐷0 and 퐷1 of 휔 . Let 푙퐴 ∶ 퐷0 → 2 and 푙퐵 ∶ 퐷1 → 2 be the characteristic functions of 퐴 and 퐵 respectively. If (퐴, 풳) ⩽W (퐵, 풴) then Player II has no winning strategy in the Borel game 퐺(푙퐴, 푙퐵) and so Player I has a winning strategy. Notice that any winning strategy 휏 for Player I in 퐺(푙퐴, 푙퐵) induces a continuous map

111 5 A Wadge hierarchy for second countable spaces

∗ 휔 휔 ∗ 휔 휏 ∶ 휔 → 휔 with the property that 휏 (훽) ∈ 퐷0 for every 훽 ∈ 휔 and if ∗ 훽 ∈ 퐷1, then 푙퐴(푓(훽)) ≠ 푙퐵(훽). Therefore when we restrict 휏 to 퐷1 we get a continuous map 푓 ∶ 퐷1 → 퐷0 such that

for every 훽 ∈ 퐷1 (푓(훽) ∉ 퐴 ⟷ 훽 ∈ 퐵), and therefore (퐵, 풴) ⩽W (풳 ∖ 퐴, 풳) as desired. One consequence of Wadge Lemma is that antichains in 퐵(2) are of size at ∁ most two. To see this, observe that (퐴, 풳) ⩽W (퐵, 풴) if and only if (퐴 , 풳) ⩽W (퐵∁, 풴). Suppose that (퐴, 풳) and (퐵, 풴) are incomparable. It follows from Wadge Lemma that that (퐵, 풴) is equivalent to (퐴∁, 풳), and therefore (퐴, 풳) is equivalent to (퐵∁, 풴). Let now (퐶, 풵) be an arbitrary element of 퐵(2). Then by ∁ Wadge Lemma again, we have either (퐴, 풳) ⩽W (퐶, 풵) or (퐶, 풵) ⩽W (퐴 , 풳).

(퐶, 풵)

⩽ W ∁ (퐴, 풳) |W (퐵, 풴) ≡W (퐴 , 풳)

⩽ W (퐶, 풵)

Figure 5.3: Consequences of Wadge Lemma

Therefore antichains are of size at most two, and up to equivalence they are necessarily of the form (퐴, 풳) and (퐴∁, 풳). As we have just seen a second consequence of Wadge Lemma is that if we stipulate that (퐴, 풳) and (퐴∁, 풳) are always equivalent, this makes 퐵(2) into a linear order, and therefore a well-order. While the continuous reducibility is ideal in the zero-dimensional framework, the situation is dramatically different in spaces like the real line for example. As far as we know the only way to prove that the continuous reducibility is well-founded on Borel subsets of the Baire space relies on two player games with perfect information. Moreover we think that the notion of dimension zero in the topological setting compares with the idea of perfect information in the game-theoretic setting. This may indicate that the restriction to zero- dimensional spaces in Theorem 5.9 is in a sense necessary. We aim to generalise the continuous reducibility – and in fact Theorem 5.9 – to spaces of higher dimension while maintaining the nice properties it enjoys

112 5 A Wadge hierarchy for second countable spaces on zero-dimensional spaces. Our first step in this direction is very simple but fundamental. We observe that the notion of function is not essential to the idea of reduction, but that on the contrary one can advantageously consider total relations in place of functions.

5.2 Reductions as total relations

The concept of reduction is used in several different fields, such as complexity theory, automata theory and descriptive set theory. While particular definitions relies on different concepts, they all share a general idea. If 풳 and 풴 are sets, 퐴 ⊆ 풳 and 퐵 ⊆ 풴, a function 푓 ∶ 풳 → 풴 is called a reduction of 퐴 to 퐵 if 퐴 = 푓 −1(퐵) or equivalently if

∀푥 ∈ 풳 (푥 ∈ 퐴 ⟷ 푓(푥) ∈ 퐵) .

Let ℱ be a class of functions from 풳 to 풳 that contains the identity on 풳 and that is closed under composition. For 퐴, 퐵 ⊆ 풳 we say that 퐴 is reducible to 퐵 with respect to ℱ if there exists 푓 ∈ ℱ such that 푓 is a reduction of 퐴 to 퐵. This always defines a quasi-order on the powerset of 풳. We now observe that as far as reducibility is concerned reductions do not need to be functions. In fact one may as well consider total relations in place of functions. We say 푅 ⊆ 풳 × 풴 is a (total) relation from 풳 to 풴, in symbols 푅 ∶ 풳 ⇉ 풴, if for all 푥 ∈ 풳 there exists 푦 ∈ 풴 with (푥, 푦) ∈ 푅. We also write 푅(푥, 푦) in place of (푥, 푦) ∈ 푅.

Definition 5.11. If 퐴 ⊆ 풳 and 퐵 ⊆ 풴 we say that a reduction of 퐴 to 퐵 is a total relation 푅 ∶ 풳 ⇉ 풴 such that

∀푥 ∈ 풳 ∀푦 ∈ 풴 [푅(푥, 푦) → (푥 ∈ 퐴 ↔ 푦 ∈ 퐵)]. (5.1)

One can also view a relation 푅 ⊆ 풳 × 풴 as the function

푅→ ∶ 풳 ⟶ 풫(풴) 푥 ⟼ 푅→(푥) = {푦 ∈ 풴 ∣ 푅(푥, 푦)}.

From this point of view, 푅 is total from 풳 to 풴 if and only if 푅→(푥) ≠ ∅ for all 푥 ∈ 풳, and (5.1) can be stated in the following way:

∀푥 ∈ 풳 [(푥 ∈ 퐴 ∧ 푅→(푥) ⊆ 퐵) ∨ (푥 ∈ 퐴∁ ∧ 푅→(푥) ⊆ 퐵∁)]. (5.2)

113 5 A Wadge hierarchy for second countable spaces

Of course, for every function 푓 ∶ 풳 → 풴, 푓 is a reduction of 퐴 to 퐵 if and only if its graph {(푥, 푓(푥)) ∣ 푥 ∈ 풳}, as a total relation from 풳 to 풴, is a reduction of 퐴 to 퐵. So our notion of reduction as total relations subsumes the notion of reduction as functions. Observe also that it follows directly from (5.2) that a total relation 푅 is a reduction of 퐴 to 퐵 if and only if it is a reduction from 퐴∁ to 퐵∁. Two total relations 푅 ∶ 풳 ⇉ 풴 and 푆 ∶ 풴 ⇉ 풵 compose to yield the total relation 푆 ∘ 푅 ∶ 풳 ⇉ 풵 in the expected way

푆 ∘ 푅 = {(푥, 푧) ∈ 풳 × 풵 ∣ ∃푦 ∈ 풴 푅(푥, 푦) ∧ 푆(푦, 푧)}.

Fact 5.12. If 퐴 ⊆ 풳, 퐵 ⊆ 풴, 퐶 ⊆ 풵, 푅 ∶ 풳 ⇉ 풵 is a reduction of 퐴 to 퐵 and 푆 ∶ 풴 ⇉ 풵 is a reduction of 퐵 to 퐶, then 푆 ∘ 푅 ∶ 풳 ⇉ 풵 is a reduction of 퐴 to 퐶. Let ℛ be a class of total relations from 풳 to 풳 that contains the diagonal {(푥, 푥) ∣ 푥 ∈ 풳} and that is closed under composition. For 퐴, 퐵 ⊆ 풳 we say that 퐴 is reducible to 퐵 with respect to ℛ if there 푅 ∈ ℛ such that 푅 is a reduction of 퐴 to 퐵. Again this defines a quasi-order on the powerset of 풳 that we call ℛ-reducibility. The following fact follows immediately from (5.1). Fact 5.13. Let 푅, 푆 ∶ 풳 ⇉ 풴, 퐴 ⊆ 풳, 퐵 ⊆ 풴. If 푅 ⊆ 푆 and 푆 is a reduction of 퐴 to 퐵, then 푅 is also a reduction of 퐴 to 퐵. Consequently, for a class ℛ as above, if we consider the the upward closure of ℛ defined by ℛ = {푆 ∶ 풳 ⇉ 풳 ∣ ∃푅 ∈ ℛ 푅 ⊆ 푆}, then the ℛ-reducibility equals the ℛ-reducibility. Therefore as far as reducibility is concerned, we gain generality by considering classes of total relations instead of classes of functions, and we can always consider classes of total relations that are upward closed. In the next section the definition of a notion of continuity for total relations between second countable 푇0 spaces is provided. We then discuss in the following sections some properties of the reducibility associated with this class of ‘continuous’ total relations.

5.3 Relatively continuous relations

We are interested in a notion of continuity for total relations called relative continuity. It relies on the concept of admissible representation of a topological

114 5 A Wadge hierarchy for second countable spaces space. While this concept is fundamental to Type-2 Theory of Effectivity (see the textbook by Weihrauch [Wei00]), we do not expect our reader to be familiar with the simple and interesting underpinning of this approach to computable analysis. We therefore review the basic definitions and provide proofs for the convenience of the reader.

5.3.1 Admissible representations A topological space 풳 is called second countable if it admits a countable base of open sets. It satisfies the separation axiom 푇0 if every two distinct points are topologically distinguishable, i.e. for any two distinct points 푥 and 푦 there is an open set which contains one of these points and not the other. It is called zero-dimensional, or 0-dimensional, if it admits a base of clopen sets, i.e. of simultaneously open and closed sets. A space is second countable and 0-dimensional if and only if it is homeomorphic to a subset of 휔휔. A Polish space is a second countable completely metrisable topological space, the Baire space is a crucial example of Polish space. Recall [Kec95, (3.11), p.17] that 0 a subspace of a Polish space is Polish if and only if it 횷2, i.e. a countable intersection of open sets. Let 풳, 풴 be second countable 푇0 spaces. If 퐴 ⊆ 풳 and 푓 ∶ 퐴 → 풴 is a function, 푓 is called a partial function from 풳 to 풴, in symbols 푓 ∶⊆ 풳 → 풴, and we refer to 퐴 as the domain of 푓, denoted by dom 푓. A partial function 푓 ∶⊆ 풳 → 풴 is continuous if it is continuous on its domain for the subspace topology on dom 푓, i.e. if for every open 푈 of 풴 there is an open 푉 of 풳 such that 푓−1(푈) = 푉 ∩ dom 푓. We quasi-order the partial functions from 휔휔 into 풳 by saying that for 푓, 푔 ∶⊆ 휔휔 → 풳

there exists a continuous ℎ ∶ dom 푓 → dom 푔 푓 ⩽rep 푔 ⟷ { with 푓(훼) = 푔 ∘ ℎ(훼) for all 훼 ∈ dom 푓.

Clearly, if 푔 is continuous and 푓 ⩽rep 푔, then 푓 is continuous too. Hence the set of partial continuous functions from 휔휔 into 풳 is downward closed with respect to ⩽rep.

Definition 5.14 ([Wei00]). Let 풳 be a topological space. A partial continuous function 휌 ∶⊆ 휔휔 → 풳 is called an admissible representation of 풳 if it is a ⩽rep-greatest element among partial continuous functions to 풳, i.e. 푓 ⩽rep 휌 holds for every partial continuous 푓 ∶⊆ 휔휔 → 풳.

115 5 A Wadge hierarchy for second countable spaces

Observe that an admissible representation 휌 of 풳 is necessarily onto 풳, since rep 휔 for every point 푥 ∈ 풳, we have 푐푥 ⩽ 휌 where 푐푥 ∶ 휔 → 풳, 훼 ↦ 푥 is the constant function. Remark 5.15. Since the subspaces of 휔휔 are up to homeomorphism the second countable 0-dimensional spaces, an admissible representation of 풳 is also a continuous map 휌 ∶ 퐷 → 풳 from some second countable 0-dimensional space 퐷 such that for every continuous map 푔 ∶ 퐸 → 풳 from a second countable 0-dimensional space 퐸 there exists a continuous map ℎ ∶ 퐸 → 퐷 such that 푔 = 휌 ∘ ℎ.

Now it is well known that every second countable 푇0 space 풳 has an admissible representation. Since it is simple and crucial for the sequel we now explain this fact in detail.

Definition 5.16. Let 풳 be a second countable 푇0 space and (푉푛)푛∈휔 be a countable base of open sets for 풳. We define the standard representation of 풳 휔 with respect to (푉푛) to be the partial map 휌 ∶⊆ 휔 → 풳 defined by

휌(훼) = 푥 ⟷ {푛 ∣ ∃푘 훼(푘) = 푛 + 1} = {푛 ∣ 푥 ∈ 푉푛}.

Notice that in the definition 휌 is indeed a function on its domain because 휔 풳 is 푇0. An 훼 ∈ 휔 codes via 휌 a point 푥 ∈ 풳 if and only if 훼 enumerates the indices of all the 푉푛’s to which 푥 belongs, while 0 can be thought of as an index for the whole space 풳 – which may not appear among the 푉푛’s.

Theorem 5.17. For every second countable 푇0 space 풳 there exists an admissible representation 휌 ∶⊆ 휔휔 → 풳. Moreover it can be chosen such that

(i) 휌 is open,

(ii) for every 푥 ∈ 풳, the fibre 휌−1(푥) is Polish.

휔 Proof. Let (푉푛) be a countable base for 풳 and let 휌 ∶⊆ 휔 → 풳 be the standard representation of 풳 with respect to (푉푛). It is enough to show that 휌 satisfies all the requirements.

−1 Continuity: Note that 휌 (푉푛) = {훼 ∈ dom 휌 ∣ ∃푘 훼(푘) = 푛 + 1} is open in 휔휔 for every 푛, so 휌 is continuous.

휔 <휔 Openness: For every basic 푁푠 = {푥 ∈ 휔 ∣ 푠 ⊆ 푥}, 푠 ∈ 휔 , we have 휌(푁푠) = ⋂ 푉 (where 푉 = 풳 by convention) which is open in 풳, so 휌 is 푘<|푠| 푠푘−1 −1 an open map.

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Polish fibres: For every point 푥 ∈ 풳

휌(훼) = 푥 ⟷ ∀푛[(∃푘 훼(푘) = 푛 + 1) ↔ 푥 ∈ 푉푛]

0 is a 횷2 definition of the fibre in 푥, so 휌 has Polish fibres. Admissibility: Let 푓 ∶⊆ 휔휔 → 풳 be continuous. Note that for every 푛 ∈ 휔 and every 훼 ∈ dom 푓

∃푘 푓(푁 ) ⊆ 푉 ⟷ 푓(훼) ∈ 푉 . 훼↾푘 푛 푛

Let 푘 ↦ ((푘)0, (푘)1) be some bijection from 휔 to 휔 × 휔 whose inverse is denoted by ⟨(푘)0, (푘)1⟩ = 푘. We define ℎ ∶ dom 푓 → dom 휌 by

(푘) + 1 if 푓(푁 ) ⊆ 푉 , 1 훼↾ (푘)1 ℎ(훼)(푘) = { (푘)0 0 otherwise.

Clearly ℎ is continuous. Moreover for every 훼 ∈ dom 푓 and every 푛 ∈ 휔, if 푓(훼) ∈ 푉 then there exists 푘 ∈ 휔 such that 푓(푁 ) ⊆ 푉 and so 푛 훼↾푘 푛 ℎ(훼)(⟨푘, 푛⟩) = 푛 + 1. Conversely, if for 훼 ∈ dom 푓 we have ℎ(훼)(푘) = 푛 + 1 then it means that 푓(푁훼↾ ) ⊆ 푉푛 and so 푓(훼) ∈ 푉푛. (푘)0 It follows that 푓(훼) = 휌 ∘ ℎ(훼) for all 훼 ∈ dom 푓, as desired.

Remark 5.18. i. If (푉푛) is a countable base of a 푇0 space 풳, the map 휎 ∶⊆ 휔휔 → 풳 defined by

휎(훼) = 푥 ⟷ Im 훼 = {푛 ∣ 푥 ∈ 푉푛}. is also an admissible representation of 풳. Indeed 휎 is continuous and for rep 휌 the standard representation of 풳 with respect to (푉푛) we have 휌 ⩽ 휎. To see this we define a monotone map 휑 ∶ 휔<휔 → 휔<휔 by induction on the length by:

휑(∅) =∅ 휑(푠) ⌢ 휑(푠)(0) if 휑(푠) ≠ ∅ 휑(푠 ⌢ 0) = { ∅ otherwise, 휑(푠 ⌢ (푛 + 1)) =휑(푠) ⌢ 푛.

This map 휑 induces a continuous map 휑∗ ∶ dom 휑∗ → 휔휔 where 휑∗(훼) = ⋃ 휑(훼↾ ) for 훼 ∈ {훼 ∈ 휔휔 ∣ lim 휑(훼↾ ) = ∞} = dom 휑∗([Kec95, 푛∈휔 푛 푛→∞ 푛

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(2.6), p.8]). We have dom 휌 ⊆ dom 휑∗ for if 훼 ∈ dom 휌 then there exists 푛 with 휌(훼) ∈ 푉푛, so there exists a 푘 minimal such that 훼(푘) > 0. Hence if 훼 = 0푘 ⌢ 훼(푘) ⌢ 훽 then 휑∗(훼)(푛) = 훼(푛 + 푘) − 1 when 훼(푛 + 푘) > 0 and 휑∗(훼)(푛) = 훼(푘) − 1 if 훼(푛 + 푘) = 0. Moreover we clearly have = 휌 = 휎 ∘ 휑∗.

ii. However, a standard representation 휌 of a second countable space 풳 relatively to some base has the following advantage. If one chooses the bijection 푘 ↦ ((푘)0, (푘)1) from 휔 to 휔×휔 such that (푘)0 ⩽ 푘 for every 푘, then the map ℎ ∶ dom 푓 → dom 휌 defined in the proof of Theorem 5.17 rep as a witness to the fact that 푓 ⩽ 휌 is actually Lipschitz, i.e. ℎ(훼)↾푛 depends only on 훼↾푛 for every 훼 ∈ dom 푓 and every 푛 ∈ 휔. This is sometimes convenient. Importantly, Brattka [Bra99, Corollary 4.4.12] showed that every Polish space 푋 has a total admissible representation, i.e. an admissible representation 휌 ∶⊆ 휔휔 → 푋 with dom 휌 = 휔휔. As an easy consequence one gets that for every second countable 푇0 space 푋: there exists an admissible representation of 푋 with a Polish domain if and only if there exists a total admissible representation of 푋. Motivated by the rich theory of Polish spaces, it is natural to consider the class of those second countable 푇0 spaces which have a total admissible representation. As a matter of fact de Brecht [deB13] showed that this class coincides with the class of quasi-Polish spaces that he recently introduced. Moreover he showed that many classical results of descriptive set theory can be generalised to this large class of non necessarily Hausdorff spaces and that the metrisable quasi-Polish spaces are exactly the Polish spaces. The real line ℝ is certainly the most important example of non zero-dimensional Polish space and we now introduce two different admissible representations for it.

Example 5.19. Let (푞푛)푛∈휔 be an enumeration of the rationals and let 퐼푛 = (푞 , 푞 ) be an enumeration of the non empty intervals of the real line ℝ with 푛0 푛1 rational endpoints. 휔 We define 휌ℝ ∶⊆ 휔 → ℝ relatively to the enumerated base (퐼푛)푛∈휔 by

휌ℝ(훼) = 푥 ⟷ Im 훼 = {푛 ∣ 푥 ∈ 퐼푛}, so that 훼 ∈ 휔휔 codes 푥 ∈ ℝ if and only if 훼 enumerates all the intervals with rational endpoints in which 푥 belongs. The second admissible representation is based on Cauchy sequences and it works mutatis mutandis for every separable complete metric space.

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Example 5.20. Let (푞푛)푛∈휔 be an enumeration of the rationals, and let 푑 be the euclidean metric on ℝ. A sequence (푥푘)푘∈휔 is said to be rapidly Cauchy if −푖 for every 푖, 푗 ∈ 휔, 푖 < 푗 implies 푑(푥푖, 푥푗) ⩽ 2 . The Cauchy representation 휔 휎ℝ ∶⊆ 휔 → ℝ of the real line is defined by

휎ℝ(훼) = 푥 ⟷ (푞훼(푘)) is rapidly Cauchy and lim 푞훼(푘) = 푥. 푘∈휔 푘→∞ This is an admissible representation of ℝ. To illustrate our ideas in the non-metrisable case we consider the Scott Do- main 풫휔, namely the powerset of 휔 partially ordered by inclusion and endowed with the Scott topology. A base of 풫휔 is given by sets of the form 푂퐹 = {푋 ⊆ 휔 ∣ 퐹 ⊆ 푋} for some finite 퐹 ⊆ 휔. This space is universal among the second countable 푇0 spaces. Indeed for every 푇0 space 풳 with some countable base (푉푛)푛∈휔 the map 푒 ∶ 풳 → 풫휔, 푥 ↦ {푛 ∣ 푥 ∈ 푉푛} is an embedding. Example 5.21. The enumeration representation of 풫휔 is the total function 휔 휌En ∶ 휔 → 풫휔 defined by

휌En(푥) = {푛 ∣ ∃푘 푥푘 = 푛 + 1}.

It is easy to see that 휌En is an open admissible representation with Polish fibres. As another example of an admissible representation of 풫휔 consider:

Example 5.22. Let (푠푛)푛∈휔 be an enumeration of the finite subsets of 휔. We 휔 define 휌<∞ ∶⊆ 휔 → 풫휔 by

휌<∞(훼) = 푥 ⟷ ∀푛 ∈ 휔 푠훼(푛) ⊆ 푠훼(푛+1) and ⋃ 푠훼(푛) = 푥. 푛∈휔

The domain of 휌<∞ is closed and 휌<∞ is clearly continuous. The map 휌<∞ is also an admissible representation of the space 풫휔 since it is continuous and rep 휔 휌En ⩽ 휌<∞, as witnessed by the continuous 푓 ∶ 휔 → dom 휌<∞ defined by

푓(훼)(푛) = 푘, where 푠푘 = {푚 ∣ ∃푗 ⩽ 푛 훼(푗) = 푚 + 1}.

5.3.2 Relative continuity The importance of admissible representations stems from the fact that continuity of a function between second countable 푇0 spaces can be accounted for “in the codes”.

119 5 A Wadge hierarchy for second countable spaces

Definition 5.23. Let 풳, 풴 be second countable 푇0 spaces. We say that a total function 푓 ∶ 풳 → 풴 is relatively continuous if for some (any) admissible representations 휌풳 and 휌풴 of 풳 and 풴 respectively, there exists a continuous 푔 ∶ dom 휌풳 → dom 휌풴, called a continuous realiser of 푓, such that 푓 ∘ 휌풳(훼) = 휌풴 ∘ 푔(훼) for ever 훼 ∈ dom 휌풳. Using the very definition of an admissible representation, it is easy tosee that a function 푓 ∶ 풳 → 풴 admits a continuous realiser for some choice of admissible representations of 풳 and 풴 if and only if it admits a continuous realiser for any choice of admissible representations.

Theorem 5.24. Let 풳, 풴 be second countable 푇0 spaces. A total function 푓 ∶ 풳 → 풴 is relatively continuous if and only if 푓 is continuous.

Proof. Let 휌풳 and 휌풴 be open admissible representations of 풳 and 풴 respectively. If 푓 ∶ 풳 → 풴 is continuous, then 푓 ∘ 휌풳 ∶ dom 휌풳 → 풴 is continuous. Since 휌풴 is admissible, there exists a partial continuous 푔 ∶ dom 휌풳 → dom 휌풴 (dom 푓 ∘ 휌풳 = dom 휌풳) with 푓 ∘ 휌풳 = 휌풴 ∘ 푔 on the domain of 휌풳, so 푓 is relatively continuous. Conversely, if 푓 ∶ 풳 → 풴 is relatively continuous there exists a partial continuous 푔 ∶ dom 휌풳 → dom 휌풴 with 푓 ∘휌풳 = 휌풴 ∘푔 on dom 푓 ∘휌풳 = dom 휌풳. Therefore 푓 ∘ 휌풳 ∶ dom 휌풳 → 풴 is continuous. So the proof will be finished once we have established the following lemma. Lemma 5.25. Let 푔 ∶⊆ 풳 → 풴 be a continuous, surjective and open map, and 푓 ∶ 풴 → 풵 any function. If 푓 ∘푔 ∶⊆ 풳 → 풵 is continuous, then 푓 is continuous. Proof. Let 푈 be open in 풵. Then 푓−1(푈) ={푔(푥) ∣ 푥 ∈ dom 푔 ∧ 푔(푝) ∈ 푓−1(푈)} since 푔 is onto, =푔((푓 ∘ 푔)−1(푈)) is open in 풴 since 푔 is an open map and 푓 ∘ 푔 is continuous.

5.3.3 Admissible representations and dimension For an admissible representation 휌 ∶⊆ 휔휔 → 풳 and a point 푥 ∈ 풳, one can think of 훼 ∈ 휔휔 with 휌(훼) = 푥 as a ‘code’ or ‘name’ for 푥. It is natural to ask what are the spaces which possess an injective admissible representation. It is actually simple to see that these spaces are exactly those of dimension zero. We now show this fact. Recall the following fact on the cardinality of a base.

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Lemma 5.26. Let 풳 be second countable. For every base 풞, there is a countable base 풞′ ⊆ 풞.

Proof. Let (푉푛) be countable base for 풳. Whenever possible choose 퐶푛,푚 ∈ 풞 with 푉푛 ⊆ 퐶푛,푚 ⊆ 푉푚. Then the countable family of the 퐶푛,푚’s is a base for 풳. Indeed for every 푥 ∈ 푉푚 there is a 퐶 ∈ 풞 with 푥 ∈ 퐶 ⊆ 푉푚 (since 풞 is a base for 풳), and furthermore there exists 푛 with 푥 ∈ 푉푛 ⊆ 퐶 ⊆ 푉푚 (since (푉푛)푛∈휔 is a base for 풳), hence 푥 ∈ 퐶푛,푚 ⊆ 푉푚.

휔 Lemma 5.27. Let 풳 be second countable 푇0 space and 휎 ∶⊆ 휔 → 풳 be an admissible representation of 풳. Then there is 퐴 ⊆ dom 휎 such that 휎↾퐴 is an open admissible representation of 풳.

Proof. Let 휌 ∶⊆ 휔휔 → 풳 be an open admissible representation of 풳 which exists by Theorem 5.17. There exists a continuous ℎ ∶ dom 휌 → dom 휎 that witnesses 휌 ⩽rep 휎. We claim that 퐴 = {ℎ(훼) ∣ 훼 ∈ dom 휌} works. Indeed rep 휔 휌 ⩽ 휎↾퐴 as ℎ also witnesses, and for every open 푂 ⊆ 휔 we have

휎↾퐴(푂) = {휎 ∘ ℎ(훼) ∣ 훼 ∈ dom 휌} = 휌(푂).

Theorem 5.28. Let 풳 be a second countable 푇0 space. The following are equivalent:

(i) 풳 is 0-dimensional,

(ii) there exists an injective admissible representation of 풳.

Proof. (i)→(ii): By Lemma 5.26, 푋 admits a countable basis (푉푛) consisting in clopen subsets of 푋, and for simplicity we may assume further that the basis is closed under complements, i.e. for every 푛 there exists 푚 with 푋 ∖ 푉푛 = 푉푚. Let 휎 ∶⊆ 휔휔 → 푋 be the partial map defined by 휎(훼) = 푥 if and only if 훼 ∶ 휔 → 2 is the characteristic function of {푛 ∈ 휔 ∣ 푥 ∈ 푉푛}. Clearly 휎 is injective and continuous. To see that 휎 is admissible, it is enough to show that 휌 ⩽rep 휎 where 휌 is the standard representation of 푋 with respect to (푉푛). This is witnessed by the continuous function 푔 ∶ dom 휌 → dom 휎 defined by

⎧1 if there exists 푘 with 훼(푘) = 푛 + 1, { 푔(훼)(푛) = ⎨0 if there exists 푘 with 훼(푘) = 푚 + 1 { ⎩ and 푉푚 = 푋 ∖ 푉푛.

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(ii)→(i): Suppose that 휌 is an injective admissible representation of 풳. It fol-

lows from Lemma 5.27 that there exists 퐴 ⊆ dom 휌 such that 휌↾퐴 is open and admissible. But since an admissible representation is surjective and 휌 is injective, we must have 퐴 = dom 휌. Therefore 휌 is a homeomorphism, and so 풳 is homeomorphic to dom 휌, hence 풳 is 0-dimensional, as desired.

5.3.4 Relatively continuous relations

We have seen that a function 푓 ∶ 풳 → 풴 between second countable 푇0 spaces is continuous if and only if it is induced by some continuous function ‘in the codes’. Moreover we have seen that when 풳 is not 0-dimensional, then no admissible representation of 풳 is injective, and so necessarily some points are to receive several codes. Since different codes of the same point can be sent onto codes of different points, a continuous function in the codes may very well induce a relation which is not functional on the spaces. Even though the resulting ‘transformations’ of the space are not necessarily functions, they are still continuous in some sense. They are called relatively continuous relations, and were first studied in [BH94].

휌풴 푦 푓(훼) = (3, 3, 2015, …)

풴

푥 푓 풳 휌풳 훼 = (3, 3, 1987, …)

Figure 5.4: Relatively continuous relation.

Definition 5.29. Let 풳 and 풴 be second countable 푇0 spaces. A total relation 푅 ∶ 풳 ⇉ 풴 is said to be relatively continuous if, for some (any) admissible

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representations 휌풳 and 휌풴 of 풳 and 풴 respectively, there exists a continuous realiser 푓 ∶ dom 휌풳 → dom 휌풴 such that for every 훼 ∈ dom 휌풳 we have

(휌풳(훼), 휌풴 ∘ 푓(훼)) ∈ 푅. Remark 5.30. Suppose 푅 ∶ 풳 ⇉ 풴 is relatively continuous with respect to 휌풳 and 휌풴 as witnessed by some continuous 푓 ∶ dom 휌풳 → 휌풴 and let 휎풳, 휎풴 rep be admissible representations of 풳 and 풴 respectively. Since 휎풳 ⩽ 휌풳 and rep 휌풴 ⩽ 휎풴 there are continuous 푔 ∶ dom 휎풳 → dom 휌풳 and ℎ ∶ dom 휌풴 → ′ dom 휎풴 with 휌풳 ∘ 푔 = 휎풳 and 휎풴 ∘ ℎ = 휌풴. Therefore if we set 푓 ∶ dom 휎풳 → ′ dom 휎풴 to be 푓 = ℎ ∘ 푓 ∘ 푔 we obtain that for every 훼 ∈ dom 휎풳 ′ 휎풴 ∘ 푓 (훼) = 휎풴 ∘ ℎ ∘ 푓 ∘ 푔(훼) = 휌풴 ∘ 푓 ∘ 푔(훼).

Now since 휎풳(훼) = 휌풳 ∘ 푔(훼) if we let 훽 = 푔(훼) we have ′ (휎풳(훼), 휎풴 ∘ 푓 (훼)) = (휌풳(훽), 휌풴 ∘ 푓(훽)) ∈ 푅, so 푅 is relatively continuous with respect to 휎풳 and 휎풴. Clearly a function 푓 ∶ 풳 → 풴 is (relatively) continuous if and only if its graph is relatively continuous as total relation from 풳 to 풴. Moreover it is easily seen that the class of relatively continuous total relations is closed under composition. Notice also that if 푅 ∶ 풳 ⇉ 풴 is relatively continuous and 푆 ∶ 풳 ⇉ 풴 is such that 푅 ⊆ 푆, then 푆 is relatively continuous too. Let 풳 and 풴 be second countable 푇0 spaces together with admissible representations 휌풳 and 휌풴. Every continuous function 푓 ∶ dom 휌풳 → dom 휌풴 induces 휌풳,휌풴 a total relation 푅푓 ∶ 풳 ⇉ 풴 defined by

휌풳,휌풴 푥 푅푓 푦 ⟷ ∃훼 ∈ dom 휌풳 (휌풳(훼) = 푥 ∧ 휌풴 ∘ 푓(훼) = 푦).

휌풳,휌풴 The function 푓 witnesses that 푅푓 is relatively continuous. In fact, 푓 wit- 휌풳,휌풴 nesses that some 푅 ∶ 풳 ⇉ 풴 is relatively continuous if and only if 푅푓 ⊆ 푅. Therefore we have the following.

Fact 5.31. Let 풳 and 풴 be second countable 푇0 and 휌풳 and 휌풴 be admissible representations of 풳 and 풴 respectively. A total relation 푅 ∶ 풳 ⇉ 풴 is relatively continuous if and only if there exists a continuous 푓 ∶ dom 휌풳 → dom 휌풴 such 휌풳,휌풴 that 푅푓 ⊆ 푅. From Theorem 5.28 and the previous fact, it follows that the relatively continuous relations from a 0-dimensional spaces are simply the continuously uni- formisable relations.

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Corollary 5.32. Let 풳 and 풴 be second countable 푇0 with 풳 0-dimensional. A total relation 푅 from 풳 to 풴 is relatively continuous if and only if it admits a continuous uniformising function, i.e. there exists a continuous 푓 ∶ 풳 → 풴 with 푅(푥, 푓(푥)) for all 푥 ∈ 풳.

It is an interesting problem to look for an intrinsic characterisation of the relatively continuous total relations, that is, one which does not rely on the notion of admissible representation. Partial answers were obtained by Brattka and Hertling [BH94] and Pauly and Ziegler [PZ13]. However, to our knowledge, the general problem is still open. We conclude this subsection with some known results in the direction. Let us say that 푅 ∶ 풳 ⇉ 풴 preserves open sets if the set

푅−1(푂) = {푥 ∈ 풳 ∣ ∃푦 ∈ 푂 푅(푥, 푦)} is open in 풳 for ever open set 푂 of 풴.

Proposition 5.33 ([BH94, Proposition 4.5]). Let 풳 and 풴 be second countable 푇0 spaces. There exists a class ℛ of total relations from 풳 to 풴 which preserves open sets such that for every 푆 ∶ 풳 ⇉ 풴

푆 is relatively continuous ⟷ ∃푅 ∈ ℛ 푅 ⊆ 푆.

Proof. Let 휌풳 and 휌풴 be admissible representations of 풳 and 풴 respectively. By Theorem 5.17 we can choose 휌풳 to be an open map. Let ℛ be the family of 휌풳,휌풴 total relations 푅푓 where 푓 ∶ dom 휌풳 → dom 휌풴 is continuous. By Fact 5.31, 휌풳,휌풴 it only remains to prove that 푅푓 preserves open sets for every continuous 푓. Indeed for every continuous 푓 ∶ dom 휌풳 → dom 휌풴 and every open 푂 of 풴

휌풳,휌풴 −1 −1 −1 (푅푓 ) (푂) = {휌풳(푥) ∣ 푥 ∈ (휌풴 ∘ 푓) (푂)} = 휌풳[(휌풴 ∘ 푓) (푂)], which is open since 휌풴 ∘ 푓 is continuous and 휌풳 is open. Moreover, in the case of a Polish codomain, Brattka and Hertling [BH94] showed the following.

Theorem 5.34. Let 풳 be second countable 푇0, 풴 be Polish, and 푅 ∶ 풳 ⇉ 풴 be such that 푅→(푥) is closed for every 푥 ∈ 풳. Then 푅 is relatively continuous if and only if there exists 푆 ∶ 풳 ⇉ 풴 that preserves open sets and such that 푆 ⊆ 푅.

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Remark 5.35. One should notice that in general preserving open sets is not a sufficient condition for the relative continuity of a total relation. Consider for example the partition of 휔휔 into

퐹 = {훼 ∈ 휔휔 ∣ ∃푛∀푘 ⩾ 푛 훼(푘) = 0} and 퐹 = 휔휔 ∖ 퐺.

Clearly 퐺 and 퐹 are both dense in 휔휔. Moreover it is well known that 퐹 ∈ 0 0 −1 휔 횺2∖횷2. Consider the total relation 푅 = (퐺×퐹 )∪(퐹 ×퐺). Then 푅 (푂) = 휔 for every non-empty open set, but 푅 not relatively continuous. Indeed any function 푓 ∶ 휔휔 → 휔휔 which uniformises 푅 needs to verify 푓−1(퐺) = 퐹 , and 0 since 퐹 is not 횷2, 푓 cannot be continuous.

5.3.5 Reduction by relatively continuous relations Aiming to generalise the continuous reducibility outside the realm of the dimension zero while maintaining its nice properties, we first observed in Sec- tion 5.2 that the notion of reduction nicely generalises from functions to total relations. We have then singled out in Subsection 5.3.4 a class of total relations between second countable 푇0 spaces which are continuous in a sense. We can now come to the definition of the notion of reducibility that we propose.

Definition 5.36. If 풳 and 풴 are second countable 푇0 spaces, 퐴 ⊆ 풳 and 퐵 ⊆ 풴, we say that 퐴 is relatively Wadge reducible to 퐵, in symbols (퐴, 풳) ≼W (퐵, 풴), if there exists a total relatively continuous 푅 ∶ 풳 ⇉ 풴 that is a reduction of 퐴 to 퐵.

We might sometimes make an abuse of notation and simply write 퐴 ≼W 퐵 in place of (퐴, 풳) ≼W (퐵, 풴) when the underlying spaces are understood. Since the class of relatively continuous relations between second countable 푇0 spaces contains the identity functions (in fact all continuous functions) and is closed under composition, ≼W is a quasi-order on the class of subsets of second countable 푇0 spaces. For every second countable 푇0 spaces 풳 and 풴, (퐴, 풳) ⩽푊 (퐵, 풴) implies (퐴, 풳) ≼W (퐵, 풴), by Theorem 5.24. However, the qo ≼W should not be con- fused with the quasi-order ⩽푊 of continuous reducibility. The quasi-order ≼W intuitively relates to topological complexity in the following sense. Assume 퐴 ⊆ 풳 and 퐵 ⊆ 풴 are subsets of second countable 푇0 spaces. Consider the problem of deciding when given a point 푥 ∈ 풳 whether 푥 belongs to 퐴 or not. Any admissible representation 휌풳 of 풳 provides an optimal way to represent this abstract problem in more concrete way, i.e. as a problem about infinite sequences of natural numbers, and therefore about

125 5 A Wadge hierarchy for second countable spaces infinite binary sequences. The membership problem in the codes for 퐴 is to decide when given a code 훼 ∈ dom 휌풳 for some point of 풳 whether 훼 is a code of a point in 퐴 or not. By the very definition of an admissible representation, the topological complexity of the membership problem in the codes for 퐴 does not depend on the chosen admissible representation. Therefore if 휌풳 and 휌풴 are admissible representations of 풳 and 풴 respectively, then (퐴, 풳) ≼W (퐵, 풴) if and only if one can continuously reduce the membership problem in the codes for 퐴 to the membership problem in the codes for 퐵. We state this observation in the following lemma which follows from Fact 5.31 and Fact 5.13.

Lemma 5.37. Let 풳 and 풴 be second countable 푇0 spaces, with admissible representations 휌풳 and 휌풴 respectively. For every 퐴 ⊆ 풳 and 퐵 ⊆ 풴 the following are equivalent

(i) (퐴, 풳) ≼W (퐵, 풴),

−1 −1 (ii) (휌풳 (퐴), dom 휌풳) ⩽푊 (휌풴 (퐵), dom 휌풴). In particular since 0-dimensional second countable spaces admits injective admissible representations, it follows that ≼W is a generalisation of ⩽W to spaces of arbitrary dimension. Proposition 5.38. Let 풳 be a 0-dimensional separable metrisable space. Then ⩽푊 and ≼W coincide on subsets of 풳. Proof. This follows from Corollary 5.32 and the previous lemma.

It is an easy consequence of the results in Section 5.1 that the quasi-order ≼W is bqo and satisfies the Wadge Lemma when suitably restricted. We however postpone this discussion a little to consider a more pressing matter. Even though we have seen that the quasi-order ≼W conveys a certain a priori idea of topological complexity, it still remains to show that it refines the Borel and Hausdorff–Kuratowski hierarchies.

5.4 Borel and Hausdorff–Kuratowski hierarchies

The classical approach initiated by the French analysts Baire, Borel, and Le- besgue to the classification of the subsets of a metric space is more descriptive in nature. Sets are classified according to the complexity of their definition from open sets. This approach was continued later by Luzin, Suslin, Haus- dorff, Sierpiński and Kuratowski.

126 5 A Wadge hierarchy for second countable spaces

As observed – apparently for the first time – by Tang [Tan79; Tan81], the classical definition of the Borel hierarchy in metric spaces is not satisfactory for non metrisable spaces. Following Selivanov [Sel06] and de Brecht [deB13] we use the following slightly modified definition of the Borel hierarchy inan arbitrary topological space (see also the paper by Spurný [Spu10]).

Definition 5.39. Let 풳 be a topological space. For each positive ordinal 훼 < 휔1 we define by induction

0 횺1(풳) = {푂 ⊆ 풳 ∣ 풳 is open},

0 ∁ 0 횺훼(풳) = {⋃ 퐵푖 ∩ 퐶푖 ∣ 퐵푖, 퐶푖 ∈ ⋃ 횺훽 for each 푖 ∈ 휔}, 푖∈휔 훽<훼 0 ∁ 0 횷훼(풳) = {퐴 ∣ 퐴 ∈ 횺훼}, 0 0 0 횫훼(풳) = 횺훼(풳) ∩ 횷훼(풳).

Proposition 5.40. For any topological space 풳 and any 훼 > 0:

0 (i) 횺훼(풳) is closed under countable union and finite intersection;

0 (ii) 횷훼(풳) is closed under countable intersection and finite union;

0 (iii) 횫훼(풳) is closed under finite union and intersection as well as under complementation.

0 0 0 Proposition 5.41. If 훼 < 훽, then 횺훼 ∪ 횷훼 ⊆ 횫훽. So the following diagram of inclusion holds between Borel classes:

횺0 횺0 횺0 1 2 ⊆ 훼 ⊆ ⊆ ⊆ ⊆ ⊆ ⊆ ⊆ ⊆ 0 0 0 0 0 횫1 횫2 횫3 ⋯ 횫훼 횫훼+1 ⋯ ⊆ ⊆ ⊆ ⊆ ⊆ ⊆ ⊆ 0 0 ⊆ 0 ⊆ 횷1 횷2 횷훼

Figure 5.5: The Borel hierarchy

Proposition 5.42. If 훼 > 2, then

0 0 횺훼(풳) = {⋃ 퐵푖 ∣ 퐵푖 ∈ ⋃ 횷훽(풳) for each 푖 ∈ 휔}. 푖∈휔 훽<훼

127 5 A Wadge hierarchy for second countable spaces

And if 풳 is metrisable the previous statement holds also for 훼 = 2, i.e.

0 0 횺2(풳) = {⋃ 퐵푖 ∣ 퐵푖 ∈ 횷1(풳) for each 푖 ∈ 휔}. 푖∈휔 Hausdorff and later Kuratowski refined the Borel hierarchy by introducing the so called difference Hierarchy. Recall that any ordinal 훼 can uniquely be expressed as 훼 = 휆 + 푛 where 휆 is limit or equal to 0, and 푛 < 휔. The ordinal 훼 is said to be even if 푛 is even, otherwise 훼 is said to be odd.

Definition 5.43. Let 휉 ⩾ 1 be a countable ordinal. For any sequence (퐶휂)휂<휉 with 훼 < 훽 < 휉 implies 퐶훼 ⊆ 퐶훽, the set 퐴 = 퐷휉((퐶휂)휂<휉) is defined by

⋃{퐶휂 ∖ ⋃ ′ 퐶휂′ ∣ 휂 odd, 휂 < 휉} for 휉 even, 퐴 = { 휂 <휂 ⋃{퐶 ∖ ⋃ 퐶 ′ ∣ 휂 even, 휂 < 휉} for 휉 odd. 휂 휂′<휂 휂

0 For a topological space 풳, 1 ⩽ 훼 < 휔1 and 1 ⩽ 휉 < 휔1 we let 퐷휉(횺훼(풳)) be the class of all sets 퐷휉((퐶휂)휂<휉) where (퐶휂)휂<휉 is an increasing sequence in 0 0 0 횺훼(풳). Notice that in particular 퐷1(횺훼) = 횺훼.

0 Of course if 푓 ∶ 풳 → 풴 is a continuous map and 퐴 ∈ 퐷휉(횺훼(풴)), then −1 0 푓 (퐴) ∈ 퐷휉(횺훼(풳)). For the Baire space, this straightforward observation is crystallised in the definition of a pointclass, that is a collection of subsets of the Baire space closed under continuous preimages, or in other words, an initial segment of the Wadge quasi-order on the Baire space. In any topological space, 0 0 0 the fact that the classes 횷훼, 횺훼 and 퐷휉(횺훼) are closed under continuous preimages means that these classes are initial segment of the quasi-order of continuous reducibility ⩽W. In this sense ⩽W refines these classical hierarchies. We now show that in an arbitrary second countable 푇0 space 풳 the classes 0 0 0 횷훼, 횺훼, 퐷휉(횺훼) enjoy the stronger and less straightforward property of being initial segments of the quasi-order ≼W. Therefore the quasi-order ≼W also refines these classical hierarchies.

Proposition 5.44. Let 풳 and 풴 be second countable 푇0 spaces and 퐴 ⊆ 풳, 퐵 ⊆ 풴. For every 1 ⩽ 훼, 휉 < 휔1,

0 0 (i) if 퐵 ∈ 횺훼(풴) and 퐴 ≼W 퐵, then 퐴 ∈ 횺훼(풳),

0 0 (ii) if 퐵 ∈ 퐷휉(횺훼(풴)) and 퐴 ≼W 퐵, then 퐴 ∈ 퐷휉(횺훼(풳)).

128 5 A Wadge hierarchy for second countable spaces

We defer the proof of the previous proposition until the end of this section since it follows from results of independent interest. The proof relies essentially on the following proposition which is a slightly modified version of a result due to Saint Raymond [Sai07, Lemma 17]. Its relevance in our context was first observed by de Brecht [deB13]. It is based on Baire category and we refer the reader to the textbook by Kechris [Kec95] for the basic definitions and results. Proposition 5.45. Let 풳 and 풴 be topological spaces, with 풳 metrisable. Let 휑 ∶ 풳 → 풴 be an open, continuous map with Polish fibres, i.e. 휑−1(푦) is Polish for all 푦 ∈ 풴. For every 푍 ⊆ 풳 define:

−1 −1 푁0(푍) = {푦 ∈ 풴 ∣ 푍 ∩ 휑 (푦) is non meagre in 휑 (푦)}, −1 −1 푁1(푍) = {푦 ∈ 풴 ∣ 푍 ∩ 휑 (푦) is comeagre in 휑 (푦)}.

Then for every positive ordinal 훼 < 휔1, 0 0 (i) If 푍 ∈ 횺훼(풳), then 푁0(푍) ∈ 횺훼(풴), 0 0 (ii) If 푍 ∈ 횷훼(풳), then 푁1(푍) ∈ 횷훼(풴). In particular, if 휑 is further assumed to be surjective then for every 퐴 ⊆ 풴 and every positive ordinal 훼 < 휔1, −1 0 0 (i) 휑 (퐴) ∈ 횺훼(풳) ⟷ 퐴 ∈ 횺훼(풴), −1 0 0 (ii) 휑 (퐴) ∈ 횷훼(풳) ⟷ 퐴 ∈ 횷훼(풴).

Proof. Since 푁1(풳 ∖ 푍) = 풴 ∖ 푁0(푍) both statements are equivalent for every 훼. Let (푉푘)푘∈휔 be a countable base for the topology of 풳. We proceed by induction on 훼. 0 For 훼 = 1 let 푍 ∈ 횺1, since 휑 is assumed to be open we have 휑(푍) is open in 풴. Since 휑−1(푦) is a Baire space for all 푦 ∈ 풴, the open subset 푍 ∩ 휑−1(푦) of −1 0 휑 (푦) is non meagre if and only if it is non empty. So 푁0(푍) = 휑(푍) ∈ 횺1(풴). So assume now that both statements are true for every 훼′ < 훼 and let 0 푍 ∈ 횺훼. Since 풳 is metrisable, 푍 is the union of a countable family (푍푛)푛∈휔 with 푍 ∈ 횷0 for some 훼 < 훼. For any point 푦 ∈ 풴, using the fact that any 푛 훼푛 푛 Borel subset of a Polish space has the Baire Property, we have the following equivalences: 푍 ∩ 휑−1(푦) is non meagre in 휑−1(푦) −1 −1 ↔ ∃푛 푍푛 ∩ 휑 (푦) is non meagre in 휑 (푦) −1 −1 ↔ ∃푛 푍푛 ∩ 휑 (푦) is comeagre in some non-empty open subset of 휑 (푦) ∁ −1 −1 −1 ↔ ∃푛 ∃푘 (푍푛 ∪ 푉푘 ) ∩ 휑 (푦) is comeagre in 휑 (푦) and 푉푘 ∩ 휑 (푦) ≠ ∅.

129 5 A Wadge hierarchy for second countable spaces

Therefore, ∁ 푁0(푍) = ⋃ 푁1(푍푛 ∪ 푉푘 ) ∩ 휑(푉푘). 푛,푘 Now 푍 ∪ 푉 ∁ ∈ 횷0 , and so 푁 (푍 ∪ 푉 ∁) ∈ 횷0 by the induction hypothesis. 푛 푘 훼푛 1 푛 푘 훼푛 0 0 Moreover 휑(푉푘) ∈ 횺1 since 휑 is an open map. It follows that 푁0(푍) is 횺훼 according to Definition 5.39. For the second claim, assume that 휑 is moreover surjective and notice that −1 if 퐴 ⊆ 풴 then for 푍 = 휑 (퐴) we have 퐴 = 푁0(푍) = 푁1(푍). Building on Proposition 5.45 and using the same technique, de Brecht [deB13] showed: Proposition 5.46. Let 풳 and 풴 be topological spaces, with 풳 metrisable. Let 휑 ∶ 풳 → 풴 be an open and continuous map with Polish fibres, 퐴 ⊆ 풴 −1 and 1 ⩽ 훼, 휉 < 휔1. If 휑 (퐴) = 퐷휉((퐶휂)휂<휉) with (퐶휂)휂<휉 an increasing 0 0 sequence in 횺훼, then 퐴 = 퐷휉(푁0(퐶휂)휂<휉). So 퐴 ∈ 퐷휉(횺훼(풴)) if and only if −1 0 휑 (퐴) ∈ 퐷휉(횺훼(풳)). Proof. Let 퐵 = 푁 (퐶 ). First let 푦 ∈ 퐴. Since 휑−1(푦) = ⋃ 퐶 ∩ 휑−1(푦) 휂 0 휂 휂<휉 휂 −1 and 휑 (푦) is Polish and non empty, there exists a least 휂푦 < 휉 such that −1 −1 −1 퐶 ∩ 휑 (푦) is non meagre in 휑 (푦), i.e. 푦 ∈ 퐵 . In particular, 퐶 ′ ∩ 휑 (푦) 휂푦 휂푦 휂 −1 ′ −1 is meagre in 휑 (푦) for all 휂 < 휂푦, hence ⋃ ′ 퐶휂′ ∩ 휑 (푦) is meagre in 휂 <휂푦 −1 −1 −1 휑 (푦). It follows that (퐶휂 ∖ ⋃ ′ 퐶휂′ ) ∩ 휑 (푦) is non meagre in 휑 (푦), 푦 휂 <휂푦 −1 so in particular it contains some 푥 ∈ 풳. Since 푥 ∈ 휑 (퐴) = 퐷휉((퐶휂)휂<휉) the parity of 휉 must differ from that of 휂푦. Therefore 푦 ∈ 퐷휉((퐵휂)휂<휉). Conversely let 푦 ∈ 퐷휉((퐵휂)휂<휉). There exists 휂푦 < 휉 whose parity is different −1 from that of 휉 such that 푦 ∈ 퐵휂 ∖⋃ ′ 퐵휂′ . Since 퐵휂 = 푁0(퐴휂), 퐶휂 ∩휑 (푦) 푦 휂 <휂푦 푦 −1 −1 −1 is non meagre in 휑 (푦), and ⋃ ′ 퐶휂′ ∩휑 (푦) is meagre in 휑 (푦). As before 휂 <휂푦 −1 −1 (퐶휂 ∖ ⋃ ′ 퐶휂′ ) ∩ 휑 (푦) is non meagre in 휑 (푦) and so in particular it 푦 휂 <휂푦 −1 must contain some point 푥 ∈ 풳. We have 푥 ∈ 퐷휉((퐶휂)휂<휉) = 휑 (퐴) and so 푦 = 휑(푥) ∈ 퐴.

Since every second countable 푇0 space has an admissible representation which is open and has Polish fibres, we obtain:

Theorem 5.47 ([deB13, Theorem 78]). Let 풳 be a second countable 푇0 space, 휌 ∶⊆ 휔휔 → 풳 an admissible representation of 풳. For any countable 훼, 휉 > 0 and every 퐴 ⊆ 풳 we have

0 −1 0 퐴 ∈ 퐷휉(횺훼(풳)) ⟷ 휌 (퐴) ∈ 퐷휉(횺훼(dom 휌)).

130 5 A Wadge hierarchy for second countable spaces

Proof. The left to right implication follows from the continuity of the admissible representation and the fact that the preimage map 휌−1 is a complete Boolean homomorphism. For the right to left implication, it is enough by Propositions 5.45 and 5.46 to show that we can assume 휌 to be open with Polish fibres – since such an admissible representation always exists by Theorem 5.17. So let 훿 ∶⊆ 휔휔 → 풳 be any admissible representation of 풳, then there exists a continuous 푓 ∶ −1 0 dom 휌 → dom 휎 with 훿∘푓 = 휌 on the domain of 휌. If 훿 (퐴) ∈ 퐷휉(횺훼(dom 훿)) then as in the first implication we have

−1 −1 −1 0 휌 (푆) = 푓 (훿 (푆)) ∈ 퐷훼(횺휃(dom 휌)). This concludes the claim. The proof of Proposition 5.44 is now straightforward.

0 0 Proof of Proposition 5.44. Since 퐷1(횺훼) is just 횺훼, (i) is a particular case of 0 (ii). Let 퐵 ∈ 퐷휉(횺훼(풴)) and suppose that 퐴 ⊆ 풳 satisfies 퐴 ≼W 퐵. Let 휌풳, 휌풴 be admissible representations of 풳, 풴 respectively. Since 퐴 ≼W 퐵, there exists a −1 −1 continuous 푓 ∶ dom 휌풳 → dom 휌풴 with (휌풴 ∘ 푓) (퐵) = 휌풳 (퐴). By continuity, −1 −1 0 휌풳 (퐴) = (휌풴 ∘ 푓) (퐵) ∈ 퐷휉(횺훼(dom 휌풳)), and so by Theorem 5.47 퐴 is 0 퐷휉(횺훼) in 풳. In the Baire space, the pointclasses and the Wadge quasi-order are two sides 0 0 0 of the same coin. Moreover while the pointclasses 횺훼, 횷훼 and 퐷휉(횺훼) are defined in terms of operations on open sets, Wadge showed thatevery non- self-dual Borel pointclass – i.e. a pointclass of Borel sets which is not closed under complementation – can be described in this fashion. Since we investigate a generalisation of the Wadge quasi-order on 휔휔 to more general spaces, it is therefore natural to wonder what is the generalisation of the notion pointclass. With Proposition 5.44 in mind and by analogy with the case of 휔휔, our approach suggests to define a ‘pointclass’ in any second countable 푇0 space as an initial segment of the quasi-order ≼W. It is worth mentioning that after the paper on which this chapter is based was finished we discovered that Louveau and Saint Raymond [LS11, Section 4] define for each Borel non-self dual pointclass Γ in 휔휔 a corresponding class Γ(풳) for every metric separable space 풳 as follows, Γ(풳) is the family of those sets 퐴 ⊆ 풳 such that for every (total) continuous map 푓 ∶ 휔휔 → 풳 we have 푓−1(퐴) ∈ Γ. It is easy to see these classes Γ(풳) are always initial 1 segments of our quasi-order ≼W. Conversely when 풳 is Polish, one can show 1Using that every Polish space has a total (i.e. defined on the whole 휔휔) admissible representation as proved by Brattka [Bra99, Corollary 4.4.12].

131 5 A Wadge hierarchy for second countable spaces

′ that if Γ is a initial segment for ≼W consisting in Borel subsets of 풳 which is not closed under complementation, then there is a Borel non-self-dual point class Γ of 휔휔 such that Γ′ = Γ(풳). However we reserve the investigation of the relation between these classes and the quasi-order ≼W as well as a general discussion on the notion of pointclass in arbitrary spaces for a later work.

5.5 A general reduction game

We now explain how the results of Section 5.1 imply that the quasi-order ≼W is bqo and satisfies the Wadge Lemma when suitably restricted. In the investigation of the quasi-order ≼W the following simple adaptation of the game in Definition 5.1 is essential.

Definition 5.48. Let 풳 and 풴 be second countable 푇0 spaces, 휌풳, 휌풴 admissible representations of 풳 and 풴 respectively, and 퐴 ⊆ 풳, 퐵 ⊆ 풴. We define a perfect information two players game 퐺휌풳,휌풴 (퐴, 퐵) as follows. Player I starts by choosing some 훼0 ∈ 휔 and then Player II chooses some 훽0 ∈ 휔, then Player I choose some 훼1 ∈ 휔, so on and so forth. Player II wins the play (훼, 훽) if and only if

훼 ∈ dom 휌풳 implies [훽 ∈ dom 휌풴 and (휌풳(훼) ∈ 퐴 ↔ 휌풴(훽) ∈ 퐵)].

휌풳 휌풳,휌풳 When 휌풳 = 휌풴 we write 퐺 (퐴, 퐵) instead of 퐺 (퐴, 퐵).

Of course the game 퐺휌풳,휌풴 (퐴, 퐵) is tightly related to the reducibility by relatively continuous relations.

Proposition 5.49. Let 풳 and 풴 be second countable 푇0 spaces, 휌풳, 휌풴 be admissible representations of 풳 and 풴 respectively. Then for all 퐴 ⊆ 풳 and 퐵 ⊆ 풴:

휌풳,휌풴 (i) If Player II has a winning strategy in 퐺 (퐴, 퐵), then 퐴 ≼W 퐵,

휌풳,휌풴 ∁ (ii) If Player I has a winning strategy in 퐺 (퐴, 퐵), then 퐵 ≼W 퐴 . Proof. A winning strategy for Player II induces a total continuous function 휔 휔 푓 ∶ 휔 → 휔 such that for every 훼 ∈ dom 휌풳, we have 푓(훼) ∈ dom 휌풴 and

휌풳(훼) ∈ 퐴 ⟷ 휌풴(푓(훼)) ∈ 퐵, or equivalently

−1 −1 훼 ∈ 휌풳 (퐴) ⟷ 푓(훼) ∈ 휌풴 (퐵).

132 5 A Wadge hierarchy for second countable spaces

Therefore if Player II has a winning strategy in 퐺휌풳,휌풴 (퐴, 퐵), then

−1 −1 (휌풳 (퐴), dom 휌풳) ⩽W (휌풴 (퐵), dom 휌풴) and so 퐴 ≼W 퐵 by Lemma 5.37. Now any winning strategy for Player I induces a continuous function 푔 ∶ 휔 휔 휔 → 휔 such that whenever 훼 ∈ dom 휌풴 then 푔(훼) ∈ dom 휌풳 and

∁ 휌풴(훼) ∈ 퐵 ⟷ 휌풳(푔(훼)) ∈ 퐴 or equivalently,

−1 −1 푔(훼) ∉ 휌풳 (퐴) ⟷ 훼 ∈ 휌풴 (퐵)

휌풳,휌풴 Therefore if Player I has a winning strategy in 퐺 (퐴, 퐵), then (퐵, 풴) ≼W (퐴∁, 풳). Remark 5.50. A priori the existence of a winning strategy for Player II in 휌풳,휌풴 퐺 (퐴, 퐵) is not equivalent to 퐴 ≼W 퐵. This is however the case when 휌풴 satisfies a stronger property than being an admissible representation. Recall that a a partial map ℎ ∶⊆ 휔휔 → 휔휔 is called Lipschitz if for every

훼, 훽 ∈ dom ℎ and every 푛 ∈ 휔, 훼↾푛 = 훽↾푛 implies ℎ(훼)↾푛 = ℎ(훽)↾푛. Let us say that a partial continuous map 휌 ∶⊆ 휔휔 → 풴 is a nice admissible representation if for every partial continuous function 푓 ∶⊆ 휔휔 → 풴 there exists a Lipschitz map ℎ ∶ dom 푓 → dom 휌 such that for every 훼 ∈ dom 푓 we have 푓(훼) = 휌(ℎ(훼)). Since Lipschitz maps are continuous, a nice admissible representation is in particular an admissible representation. Moreover, as we noticed in Remark 5.18 ii., the standard representation of a second countable 푇0 space 풴 relatively to some base is always a nice admissible representation of 풴. Hence every second countable 푇0 space has a nice admissible representation. Now if 휌풴 is a nice admissible representation of 풴, then Player II has a 휌풳,휌풴 winning strategy in 퐺 (퐴, 퐵) if and only if 퐴 ≼W 퐵. Indeed if 퐴 ≼W 퐵, then there exists some relatively continuous reduction of 퐴 to 퐵, and therefore ′ there exists some continuous function 퐹 ∶ dom 휌풳 → dom 휌풴. Since 휌풴 is a nice admissible representation there exists a Lipschitz map 퐹 ∶ dom 휌 → dom 풳 휌풴 ′ such that 휌풴 ∘ 퐹 = 휌풴 ∘ 퐹 . This means that 퐹 (훼)↾푛 depends only on 훼↾푛 for 휌풳,휌풴 every 훼 ∈ dom 휌풳. Therefore a winning strategy for Player II in 퐺 (퐴, 퐵) is for example given by the function 휎 ∶ 휔<휔 → 휔<휔 defined by induction on

133 5 A Wadge hierarchy for second countable spaces the length by:

휎(∅) = ∅ ⎧퐹 (훼)↾ for some 훼 ∈ 푁 ∩ dom 휌 , if any, { |푠| 푠 풳 휎(푠) = 휎(푠↾ ) ⌢ 0|푠|−푘 if 푁 ∩ dom 휌 = ∅ and 푘 < |푠| is the largest ⎨ 푘 푠 풳 { such that 푁 ∩ dom 휌 ≠ ∅. ⎩ 푠↾푘 풳 휔 Observe that when dom 휌풳 and dom 휌풴 are Borel subsets of 휔 , the reduction game 퐺휌풳,휌풴 (퐴, 퐵) is Borel and therefore determined by Martin’s result as long as 퐴 ⊆ 풳, 퐵 ⊆ 풴 are Borel subsets of 풳 and 풴 respectively. We are therefore naturally led to the following definition.

Definition 5.51. A second countable 푇0 space 풳 is called Borel representable if there exists an admissible representation 휌 ∶⊆ 휔휔 → 풳 of 풳 such that dom 휌 Borel in 휔휔. It is clear that every Borel subset of the Scott domain 풫휔 is Borel representable. We do not know if the converse is true, namely: Problem 3. Is every Borel representable space homeomorphic to a Borel subset of 풫휔? From Proposition 5.49 and Borel determinacy we obtain the following generalisation of Lemma 5.10. Theorem 5.52 (Generalised Wadge Lemma). Let 풳 and 풴 be a Borel representable spaces, 퐴 ⊆ 풳 and 퐵 ⊆ 풴 be Borel. Then

∁ either (퐴, 풳) ≼W (퐵, 풴) or (퐵, 풴) ≼W (퐴 , 풳). For the same reasons already exposed after Lemma 5.10, the quasi-order ≼W has a very simple structure when restricted to Borel subsets of Borel representable spaces. Namely antichains are of size at most two, and up to equivalence they are necessarily of the form (퐴, 풳) and (퐴∁, 풳). Moreover if ∁ we stipulate that (퐴, 풳) and (퐴 , 풳) are always equivalent, this makes ≼W into a linear order. Using the above game, the well-foundedness of ≼W on Borel subsets of Borel representable spaces can be obtained by following the proof of the well- foundedness of the Wadge quasi-order on 휔휔 as for example presented by Kechris [Kec95, p. 21.15]. However we think it is best seen as a particular case of the following extension of Theorem 5.9 to Borel representable spaces. Let us first extend Definition 5.8 in the obvious way:

134 5 A Wadge hierarchy for second countable spaces

Definition 5.53. Let 푄 be a quasi-order. A map 푙 ∶ 풳 → 푄 from some Borel representable space 풳 is called a Borel 푄-labelling function of 풳 if

(1) for every 푞 ∈ 푄, the set 푙−1(푞) is Borel in 풳,

(2) the image Im 푙 of 푙 is countable.

We refer to 풳 as the domain of 푙 and denote it by dom 푙. We let 퐵gen(푄) be the class of all Borel 푄-labelling of some Borel representable space. We quasi-order 퐵gen(푄) by

푙0 ≼c 푙1 ⟷ there exists a relatively continuous 푅 ∶ dom 푙0 ⇉ dom 푙1

such that for every 푥 ∈ dom 푙0 and every 푦 ∈ dom 푙1,

푥 푅 푦 implies 푙0(푥) ⩽ 푙1(푔(푥)) in 푄.

We now have the following straightforward generalisation of Theorem 5.9.

Theorem 5.54. If 푄 is bqo, then 퐵gen(푄) is bqo.

Proof. Let ℎ ∶ [휔]∞ → 퐵gen(푄) be a locally constant multi-sequence. We can choose for every 푙 ∈ 퐵gen(푄) in the countable image of ℎ an admissible 휔 representation 휌dom 푙 ∶ 퐷푙 → dom푙 with 퐷푙 a Borel subset of 휔 . We get a ̃ ∞ 퐵 ̃ multi-sequence ℎ ∶ [휔] → 픏 (푄) by letting ℎ(푋) = ℎ(푋) ∘ 휌dom ℎ(푋). Since 퐵 ∞ ̃ 픏 (푄) is bqo by Theorem 5.7, there exists 푋 ∈ [휔] such that ℎ(푋) ⩽픏 ̃ ℎ(∗푋). To lighten the notation, let ℎ(푋) = 푙0 ∶ 풳0 → 푄 and ℎ(∗푋) = 푙1 ∶ 풳 → 푄 so that we have 푙 ∘ 휌 ⩽ 푙 ∘ 휌 , i.e. Player II has a winning 1 0 풳0 픏 1 풳1 strategy 휎 in 퐺(푙 ∘ 휌 , 푙 ∘ 휌 ). Now this winning strategy for Player II 0 풳0 1 풳1 induces a continuous function 휎∗ ∶ 휔휔 → 휔휔 which restricts to a continuous 푓 ∶ dom 휌 → dom 휌 such that for every 훼 ∈ dom 휌 we have 풳0 풳1 풳0 푙 ∘ 휌 (훼) ⩽ 푙 ∘ 휌 (푓(훼)) in 푄. 0 풳0 1 풳1 Therefore the continuous function dom 휌 → dom 휌 induces a relatively 풳0 풳1 continuous total relation 푅 ∶ 풳0 → 풳1 that witnesses 푙0 ≼c 푙1, i.e. ℎ(푋) ≼c gen ℎ(∗푋), and so ℎ is good. Since ℎ was arbitrary, it follows that 퐵 (푄) is bqo. When 푄 is the bqo 2 partially ordered by equality, then 퐵gen(2) consists of the Borel subsets of Borel representable spaces quasi-ordered by ≼W.

Corollary 5.55. The quasi-order ≼W is well-founded on the Borel subsets of Borel representable spaces.

135 5 A Wadge hierarchy for second countable spaces

Of course assuming the Axiom of Determinacy (AD), the general structural result holds. Namely assuming AD, all subsets of any second countable 푇0 space 풳 are bqo under ≼W, and if 퐴 ⊆ 풳 and 퐵 ⊆ 풴 are subsets of second ∁ countable 푇0 spaces, then either (퐴, 풳) ≼W (퐵, 풴), or (퐵 , 풴) ≼W (퐴, 풳). These positive results on the structure of the quasi-order ≼W also imply that ≼W often differs with the quasi-order of continuous reducibility ⩽W. Indeed Schlicht [Sch] showed that in every non 0-dimensional metric space there exists an antichain of the size of the continuum for the continuous reducibility. Using this result and Proposition 5.38, we see that in the separable metrisable case the two notions of reducibility differ as soon as we depart from the zero-dimensional framework.

Corollary 5.56. Let 풳 be a metrisable and Borel representable space. Then ≼W and ⩽W coincide on subsets of 풳 if and only if 풳 is 0-dimensional.

5.6 The real line and the Scott domain

We now briefly have a closer look at the difference between the continuous reducibility and the reducibility by continuous relations in two important examples.

5.6.1 The Real Line

In [Ike10] (see also [IST]) an embedding from (풫(휔), ⊆fin), where ⊆fin denotes inclusion modulo finite, i.e. 푥 ⊆fin 푦 ↔ 푥 ∖ 푦 is finite, into the difference oftwo open sets of the real line ordered by Wadge reducibility is exhibited. We now recall this construction. 휔 Take increasing sequences of real numbers ⟨푎훼, 푏훼 ∣ 훼 < 휔 ⟩ indexed by the 휔 ordinal 휔 and ⟨푐푛 ∣ 푛 ⩾ 1⟩ with

휔 푎훼 < 푏훼 < 푎훼+1 for each 훼 < 휔 − 휔 푎휆 ∶= sup{푎훼 ∣ 훼 < 휆} < 푎휆 for each limit 휆 < 휔 − 푎휔푛 < 푐푛 < 푎휔푛 for each 푛 ∈ 휔.

Now for 푋 ⊆ 휔 ∖ {0} we let

퐷푋 = ⋃ [푎훼, 푏훼) ∪ {푐푛 ∣ 푛 ∉ 푋}. 훼<휔휔

Clearly 퐷푋 is a difference of two open sets for all 푋 ⊆ 휔 ∖ {0}.

136 5 A Wadge hierarchy for second countable spaces

Theorem 5.57 ([Ike10]). For every 푋, 푌 ⊆ 휔 ∖ {0},

푋 ⊆fin 푌 ⟷ 퐷푋 ⩽푊 퐷푌 .

By Parovičenko’s Theorem [Par93], any poset of size ℵ1 embeds into the partially ordered set (풫(휔), ⊆fin), hence there are long infinite descending chains and long antichains for the Wadge reducibility, already among the difference of two open sets of the real line. As an example, we now give winning strategies witnessing 퐷푋 ≼W 퐷푌 for every 푋, 푌 ⊆ 휔 ∖ {0}.

Proposition 5.58. For every 푋, 푌 ⊆ 휔 ∖ {0}, we have 퐷푋 ≼W 퐷푌 .

Proof. Let 휌ℝ be the admissible representation of the real line from Example 5.19. 푥 We choose for every 푥 ∈ ℝ a particular code via 휌ℝ by setting 훼 ∶ 휔 → 휔 to be the increasing enumeration of {푛 ∈ 휔 ∣ 푥 ∈ 퐼푛}. Now fix 푋, 푌 ⊆ 휔 ∖ {0}. We describe a winning strategy 휎 = 휎푋,푌 for 휌ℝ − player II in the game 퐺 (퐷푋, 퐷푌 ). Let 퐽푘 be the open interval (푎휔푘 , 푎휔푘 ). And note that we only need to consider positions where Player I has played (푛 , 푛 , … , 푛 ) with ⋂푗 퐼 is non empty. Let 푋 △ 푌 denote the symmetric 0 1 푗 푖=0 푛푖 difference of 푋 and 푌 , i.e.

푋 △ 푌 = {푥 ∈ 휔 ∖ {0} ∣ ¬(푥 ∈ 푋 ↔ 푥 ∈ 푌 )}.

휔 휌ℝ Our winning strategy 휎 ∶ 휔 → 휔 for Player II in 퐺 (퐷푋, 퐷푌 ) goes as follows: As long as Player I is in a position where he has played (푛0, 푛1, ⋯ , 푛푗) such that 퐼푗 = ⋂푗 퐼 ⊈ 퐽 for all 푘 ∈ 푋 △ 푌 , 휎 consists simply in copying the 푖=0 푛푖 푘 last move of Player I: 푛푗. Therefore 휎 will induce the identity function outside the 퐽푘’s for which 푘 ∉ 푋 △ 푌 . Now consider Player I has played (푛0, 푛1, ⋯ , 푛푗) such that there exists 푘 ∈ 푋 △푌 with 퐼푗 = ⋂푗 ⊆ 퐽 and let 푙 be the least integer with 퐼푙 = ⋂푙 퐼 ⊆ 푖=0 푘 푖=0 푛푖 퐽푘. We distinguish several cases:

푐푘 ∈ 퐷푌 ∖ 퐷푋: then for 휎 to be winning for Player II, it must eventually make him play the code of a point outside of 퐷푌 and it cannot be 푐푘. 푙 푙 Now since 퐼 ⊆ 퐽푘, say 퐼 = (푟0, 푟1), we can for example choose 푟 + min{푟 , 푐 } 푦 = 0 1 푘 , if 푟 < 푐 , 2 0 푘 max{푟 , 푐 } + 푟 or 푦 = 0 푘 1 , if 푐 ⩽ 푟 , 2 푘 0

137 5 A Wadge hierarchy for second countable spaces

and play 훼푦(푗 − 푙).

In other words, if Player I enters some 퐽푘 with 푐푘 ∈ 퐷푌 ∖ 퐷푋, then 휎 consists in playing the code of some 푦 ∈ 퐽푘 different from 푐푘, where 푦 depends on the first position where Player I enters 퐽푘.

푗 푗 푐푘 ∈ 퐷푋 ∖ 퐷푌 and 푐푘 ∈ 퐼 : then as long as 푐푘 ∈ 퐼 , 휎 must consist in playing as if Player I was going to play 푐푘, i.e. describe step by step a point belonging to 퐷푌 and it cannot be 푐푘. 푙−1 푙−1 Now since 퐼 ⊈ 퐽푘 (if 푙 = 0 set 퐼 = ℝ), we choose some 푦 ∈ 퐷푌 ∩퐼푙−1 as follows: 푙−1 • if 푎휔푘 ∈ 퐼 , then set 푦 = 푎휔푘 , 푘 푙−1 • otherwise there is a minimal 훽 < 휔 with 푎훽 ∈ 퐼 , set 푦 = 푎훽, and we play 훼푦(푗 − 푙).

푗 푐푘 ∈ 퐷푋 ∖ 퐷푌 and 푐푘 ∉ 퐼 : then for 휎 to be winning for Player II, it must eventually make him play the code of a point which is outside of 퐷푌 , but we must be careful to be consistent with what Player II has already played until that point. 푘 Let 푝 be the least integer such that 푐푘 ∉ 퐼 . First if 푝 ⩽ 푙, i.e. at the first position where Player I entered 퐽푘 we already knew he was not going to 푙 play 푐푘, so we can just copy its last move 푛푗. Otherwise 푙 < 푝 so 푐푘 ∈ 퐼 and we must distinguish two cases: 푙−1 • if 푎휔푘 ∈ 퐼 , then according to our previous case, at round 푝, Player II has so far played according to 휎:

푎 푘 푎 푘 푎 푘 푡 = (푛0, 푛1, … , 푛푙−1, 훼 휔 (0), 훼 휔 (1), … , 훼 휔 (푝 − 푙 − 1)).

so ⋂푝−1 퐼 is an open interval (푟 , 푟 ) with rational endpoints 푖=0 푡(푖) 0 1 satisfying 푟0 < 푎휔푘 < 푟1, so we can take

− max{푎 , 푟 } + 푎 푘 푧 = 휔푘 0 휔 2 and play 훼푧(푗 − 푝). • Otherwise according to our previous case, up to round 푝, the moves of Player II according to 휎 are

푎훽 푎훽 푎훽 푡 = (푛0, 푛1, … , 푛푙−1, 훼 (0), 훼 (1), … , 훼 (푝 − 푙 − 1)).

138 5 A Wadge hierarchy for second countable spaces

where 훽 is the minimal ordinal with 푎 ∈ 퐼푙−1. Again ⋂푝−1 퐼 훽 푖=0 푡(푖) is an open interval (푟0, 푟1) with rational endpoints satisfying 푟0 < 푎훽 < 푟1, so we can take

max{푎−, 푟 } + 푎 푧 = 훽 0 훽 2

− 푧 where 푎훽 stands for 푏훽−1 if 훽 is successor, and we play 훼 (푗 − 푝).

휌ℝ It should be clear that 휎 is a winning strategy for Player II in 퐺 (퐷푋, 퐷푌 ). So 퐷푋 ≼W 퐷푌 . ⩽̸ If 푋 ⊈fin 푌 , then 푋 W 푌 by Theorem 5.57 and so the winning strategy 휌ℝ for II in 퐺 (퐷푋, 퐷푌 ) described in the previous proof induces a continuous 휔 휔 푓푋,푌 ∶ 휔 → 휔 . The relation

푅휌ℝ (푥, 푦) ⟷ ∃훼 ∈ dom 휌 (휌 (훼) = 푥 ∧ 푦 = 휌 (푓 (훼))) 푓푋,푌 ℝ ℝ ℝ 푋,푌 is therefore a relatively continuous relation from ℝ to ℝ with no continuous uniformising function. Indeed any function uniformising 푅휌ℝ is a reduction 푓푋,푌 ⩽̸ of 푋 to 푌 and since 퐷푋 W 퐷푌 there is no such continuous function.

5.6.2 The Scott Domain

We now give a simple example in the space 풫휔 of a case where ⩽W differs from ⩽̸ ≼W. Consider {{0}}, {휔} ⊆ 풫휔, we first show that {{0}} W {휔}. To see this, recall that continuous functions on 풫휔 are the Scott continuous functions with respect to inclusion, so in particular they are monotone for inclusion. Now since 휔 is the top element, any monotone map 푓 ∶ 풫휔 → 풫휔 with 푓({0}) = 휔 has −1 to send every 푋 ⊆ 휔 with 0 ∈ 푋 onto 휔 too, so that 푓 (휔) ⊇ 푂{0}. Therefore no Scott continuous function is a reduction from {{0}} to {휔}. ⩽̸ While we have {{0}} W {휔}, we actually have {{0}} ≼W {휔}, i.e. there exists a relatively continuous 푅 ∶ 풫휔 ⇉ 풫휔 such that for all 푋, 푌 ∈ 풫휔:

푋 푅 푌 ⟶ (푋 = {0} ↔ 푌 = 휔).

Clearly any such relation 푅 cannot be uniformised by a Scott continuous function. Indeed such a Scott continuous function would be a reduction between the considered sets, and we know there is none.

139 5 A Wadge hierarchy for second countable spaces

We now give such a relation 푅 as a strategy in the game 퐺휌En ({{0}}, {휔}). Since 휌En is total, we know by Lemma 5.37 that {{0}} ≼W {휔} if and only if 퐴 ⩽푊 퐵 for 퐴 = 휌−1({{0}}) = {훼 ∈ 휔휔 ∣ 훼 ∈ 2휔 ∧ ∃푘 훼(푘) = 1} and 퐵 = 휌−1({휔}) = {훼 ∈ 휔휔 ∣ 훼 ∶ 휔 → 휔 is surjective}. A winning strategy for Player II is for example given by the function 휎 ∶ 휔<휔 → 휔 defined by

<휔 0 if 푠 ∈ {0} or ∃푘 < |푠| 푠푘 ≠ 0, 1, 휎(푠) = { 휔 푛 if 푠 ∈ 2 and 푛 = |푠| − min{푘 ∣ 푠푘 = 1}. It is easily seen that this strategy induces a continuous function 푓 ∶ 휔휔 → 휔휔 witnessing the relative continuity of the relation 푅 ∶ 풫휔 ⇉ 풫휔 given by 푅({0}) = {휔}, 푅(푋) = {∅} if 0 ∉ 푋, 푅(푋) = {푛 ∣ 푛 ∈ 휔} if {0} ⊂ 푋, where 푛 = {0, … , 푛 − 1} and ⊂ denotes strict inclusion. We close this chapter by mentioning another crucial difference between ≼W and ⩽W in relation with complete sets. Recall (e.g. [Kec95]) that 퐹 = {훼 ∈ 휔 0 휔 0 휔 휔 ∣ ∃푛∀푘 ⩾ 푛 훼(푘) = 0} is complete for 횺2(휔 ), i.e. 퐹 ∈ 횺2(휔 ) and for 0 휔 휔 every 퐴 ∈ 횺2(휔 ) we have 퐴 ⩽W 퐹 . In fact a subset of 휔 is complete for 0 휔 0 휔 0 휔 횺2(휔 ) if and only if it belongs to 횺2(휔 ) ∖ 횷2(휔 ). 0 The set 풫<∞(휔) of finite subsets of 휔 is 횺2 in 풫휔. It is shown in [BG15, Theorem 5.10] that it is not complete for the Scott continuous reducibility in 0 0 ⩽̸ the class 횺2(풫휔), i.e. there exists 퐺 ∈ 횺2(풫휔) such that 퐺 W 풫<∞(휔). In contrast the following easy result holds. 0 Lemma 5.59. We have 횺2(풫휔) = {퐴 ⊆ 풫휔 ∣ 퐴 ≼W 풫<∞(휔)}. 0 Proof. Since 풫<∞(휔) is 횺2 in 풫휔, the right to left inclusion follows from Proposition 5.44. 휔 Now for the admissible representation 휌En ∶ 휔 → 풫휔 from Example 5.21, we have ̃ −1 휔 퐹 = 휌En(풫<∞(휔)) = {훼 ∈ 휔 ∣ ∃푛∀푘 훼(푘) ⩽ 푛}. ̃ 휔 휔 Clearly 퐹 ⩽W 퐹 as the continuous function 푓 ∶ 휔 → 휔 , 푓(훼)(푛) = Card{푘 < ̃ 0 휔 푛 ∣ 훼(푘) ≠ 0} witnesses. Hence 퐹 is also complete for 횺2 in 휔 . Therefore for 0 휔 휔 any 횺2 set 퐴 ⊆ 풫휔, there is a continuous function 푓 ∶ 휔 → 휔 which reduces −1 −1 휌En(퐴) to 휌En(풫<∞(휔)), and so 퐴 ≼W 풫<∞(휔).

140 5 A Wadge hierarchy for second countable spaces

We end here our brief exemplification of the difference between the quasi- orders ⩽W and ≼W on the real line and the Scott domain. However, the study of the reducibility by relatively continuous relations on these two spaces, and others, yet remains to be realised.

141 6 Conclusion

The notion of better-quasi-order

One aim throughout this thesis was to understand better the notion of bqo. We have have committed ourselves to motivate as best as we could the definition of bqo in Chapter 2. Nonetheless, we could not help but to notice that the way we arrived to the definition was somehow accidental. We therefore formulated the following problem (see Problem 1):

Problem 4. Characterise the topological digraphs which can be substituted for ([휔]∞, 햲) in the definition of bqo.

In a first attempt to apprehend this problem, we proved Theorem 2.69 which shows that one can at least substitute a so called generalised shift for the shift map 햲. At present, a complete solution however seems elusive.

Fronts as uniform spaces

In Chapter 3 we investigated fronts as uniform spaces. We believe that our results, especially Theorems 3.24 and 3.25, entertain the relevance of this approach and we hope that these ideas can be applied in other contexts where fronts are used. In any case, these techniques allow for a smooth proof of Pouzet’s conjecture as presented in Chapter 4.

Unravel the notion of better-quasi-order

One obstacle in understanding the notion of bqo may lie in the difficulty to appreciate the gap between being wqo and being bqo. In this regard our Theorem 4.40 (Pouzet’s conjecture) provides a first step in understanding the notion of bqo. Notice that it provides us with a process which can be iterated. Suppose 푄 is wqo, and consider Id*(푄). If Id*(푄) is not wqo, then 푄 is not bqo. But assume now that Id*(푄) is wqo, we can apply Theorem 4.40 once again and can consider Id*(Id*(푄)) = Id2(푄). If Id2(푄) is not wqo, then 푄 is

142 6 Conclusion not bqo, otherwise we can consider Id*(Id2(푄)) = Id3(푄), so on and so forth. There are however examples of wqos such that Id푛(푄) is wqo for ever natural number 푛, while 푄 admits a bad super-sequence from the Schreier barrier. We believe that finding a way to go on unravelling a wqo into the transfinite could give new insights into the notion of bqo. As an example of a related problem we cannot answer, we would like to mention the following challenging question (see [Pou78]): Problem 5 (R. Bonnet). Is every wqo poset a countable union of bqo posets?

Better-quasi-ordering the subsets of a topological space

In Chapter 5 we have defined a quasi-order on the subsets of an arbitrary second countable topological 푇0 space by considering reductions by relatively continuous relations. We think it conveys a natural idea of relative complexity and we have proved that it refines the classical Borel and Hausdorff-Kuratowski hierarchies. Moreover we showed that, contrary to the quasi-order of reducibility by continuous functions, it is bqo on the Borel subsets of every space belonging to the very large class of Borel representable spaces. However, this quasi-order has not yet been really studied on alternative spaces, not even on Polish spaces. To give an indication of what could be expected, it is worth mentioning that Duparc and Fournier [DF] have studied the quasi-order of reducibility by relatively continuous relations on the space 휔⩽휔 of finite or infinite sequences of natural numbers, equipped withthe prefix topology1. They showed that for every Borel non-self-dual subset 퐴 of 휔휔 – i.e. 휔 ⩽̸ ∁ ′ a Borel subset 퐴 of 휔 such that 퐴 W 퐴 – there exists a Borel subset 퐴 of ⩽휔 ′ ′ ′ 휔 such that 퐴 ≼W 퐴 and 퐴 ≼W 퐴, and moreover every Borel subset 퐴 of 휔⩽휔 appears in this way. This result motivates the following general question: Problem 6. Let 풳 be an uncountable quasi-Polish space. Is it true that for every non-self-dual Borel subset 퐴 of 휔휔 there exists a subset 퐴′ of 풳 with ′ ′ 퐴 ≼W 퐴 and 퐴 ≼W 퐴? In a different but related direction, we recall that one very satisfactory fact about the Wadge quasi-order on the Baire space is crystallised in the following observation: 1Namely the Scott topology associated with the prefix relation, a base for which is given by the sets of the form {푢 ∈ 휔⩽휔 ∣ 푠 ⊑ 푢} for some 푠 ∈ 휔<휔.

143 6 Conclusion

Thus we have come to a full circle – non-self-dual pointclasses considered by early descriptive set theorists were defined in terms of (explicit) operations, and assuming AD every non-self-dual pointclass is defined in terms of operations on open sets. Andretta and Louveau [AL, p.5] Now if 풳 is a quasi-Polish space, let us say that a family Γ of subsets of 풳 is a non-self-dual initial segment for ≼W if Γ is not closed under complementation and, 퐴 ≼W 퐵 and 퐵 ∈ Γ implies 퐴 ∈ Γ. We believe that the following question is worth studying.

Problem 7. Assume AD and let 풳 be a quasi-Polish space. Is every non-self- dual initial segment for ≼W defined in terms of operations on open sets?

144 List of Symbols

Below are listed the main symbols and abbreviations used throughout the thesis together with a short description and the numbers of the pages where they are introduced.

푋/푠, 푋/푛 {푘 ∈ 푋 ∣ 푘 > max 푠}, {푘 ∈ 푋 ∣ 푘 > 푛} 21, 28 [푋]푘 {푠 ⊆ 푋 ∣ |푠| = 푘} 28 푆↑ the set of upper bounds of 푆 78 ⋃ 퐹 {푛 ∣ ∃푠 ∈ 퐹 푛 ∈ 푠} 30 ↓ 푆, ↓ 푞 {푝 ∣ ∃푠 ∈ 푆 푝 ⩽ 푠}, {푝 ∣ 푝 ⩽ 푞} 19 푃̂ the profinite completion of the po 푃 88 [푋]∞ the infinite subsets of 푋 28 휔op the opposite of the po (휔, ⩽) 73 ⊑, ⊏ 푠 ⊏ 푡 ↔ ∃푘 ∈ 푡 푠 = {푛 ∈ 푡 ∣ 푛 < 푘} 30, 31 ∗푋 the shift 푋 ∖ {min 푋} 41 ¡ ¡ ∞ 푠 푡 ↔ ∃푋 ∈ [휔] (푠 ⊏ 푋 ∧ 푡 ⊏ ∗푋) 41 푝 | 푞 푝 ⩽̸ 푞 and 푞 ⩽̸ 푝 15 |푠| the cardinality of 푠 28 퐵(푄) the qo of Borel 푄-labelling functions of 0-dim Luzin spaces 111 퐵gen(푄) the qo of Borel 푄-labelling functions of Borel representable spaces 135 bqo better-quasi-order 41 풟(푄) downsets of 푄 with inclusion 19 풟fb(푄) finitely bounded downsets of 푄 with inclusion 19 헗(퐿) the Priestley dual of the lattice 퐿 82 0 횫훼 the ambiguous class 훼 127 0 0 퐷휉(횺훼) the class of 휉-differences of 횺훼 128 험(풳) the lattice dual to the Priestley space 풳 82 ∗ 퐺푉 ∗ the game defining the qo on 푉 (푄) 38 퐵 퐺(푙0, 푙1) the game defining the qo on 픏 (푄) 104 퐺휌풳,휌풴 (퐴, 퐵) the reduction game relatively to the representations 132 퐻∗ 푄 qo of hereditarily countable non-emptys sets over 푄 45 휔1 Id(푃 ) the ideal completion of the po 푃 , the ideal space of the wqo 푃 72, 80 Id*(푃 ) the po of non principal ideals of 푃 73

145 List of Symbols

픏퐵(푄) the qo of Borel 푄-labelling functions of Borel subsets of 휔휔 104 ⩽rep the quasi-order on partial functions 푓 ∶⊆ 휔휔 → 풳 115 CId(푃 ) the Cauchy ideal completion of 푃 76 푉 ∗(푄) qo of non-empty sets over 푄 38 풫 (푄) qo of countable subsets of 푄 19 <ℵ1 풫(푄) powerset of the qo 푄 19 풮 the Schreier barrier 29 Im 푓 the image or range of the function 푓 48 0 횷훼 the multiplicative class 훼 127 풫휔 the Scott domain 119 ℜ Rado’s partial order 25 휌En an admissible representation of the Scott domain 119 휌<∞ another admissible representation of the Scott domain 119 휌ℝ an admissible representation of the real line 118 seq푛∈푋 퐹푛 {{푛} ∪ 푠 ∣ 푛 ∈ 푋 and 푠 ∈ 퐹푛} 28 0 횺훼 the additive class 훼 127 휎ℝ an admissible representation of the real line based on Cauchy sequences 119 TailLat(푃 ) the lattice lattice of the po 푃 85 ≼W reducibility by relatively continuous relations 125 wqo well-quasi-order 18

146 Index

Below are listed the main definitions and theorems appearing in this thesis together with the numbers of the pages where they are introduced and discussed. better-quasi-order (bqo) 41, 42, 47 better-relation 47 block 58 Borel hierarchy 127 Borel representable space 134 canonical extension 92 Cantor-Bendixson rank 55 Cauchy ideal 76 difference hierarchy 128 downset 19 finite basis property 19 finitely bounded 19 front 28, 30 sub-front 33 ideal 72 principal ideal 72 ideal completion 72 ideal space of a wqo 80 initial segment 19 interval order 98

Lawson topology 75, 79 multi-sequence 35 bad multi-sequence 41 Borel multi-sequence 106 locally constant multi-sequence 36 perfect multi-sequence 47

147 Index

Nash-Williams’ theorem 34 normal 67 partial order (po) 16 Priestley duality 82 profinite completion of a partial order 88 property M 78, 79 quasi-order (qo) 15

Rado’s partial order 25, 94 Rado’s trick 23 ray 30 reduction 113 regular sequence 21 relatively continuous relatively continuous function 120 relatively continuous relation 122 relatively Wadge reducible 125 representation admissible representation 115 standard representation 116

Schreier barrier 29 super-sequence 29 bad super-sequence 41 Cauchy super-sequence 54, 56, 65 perfect super-sequence 47 spare super-sequence 37 sub-super-sequence 33 tail lattice 85 topological digraph 46 tree 31 uniformly continuous 52, 59 upset 19 well-founded 18 well-quasi-order (wqo) 18

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