Zero-Dimensional $\Sigma $-Homogeneous Spaces

$Zero-Dimensional $\Sigma $-Homogeneous Spaces$

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By space we will always mean separable metrizable topological space. A space X is homogeneous if for every x, y P X there exists a homeomorphism h : X ÝÑ X such that hpxq“ y. For example, using translations, it is easy to see that every topological group is homogeneous (as [vE2, Corollary 3.6.6] shows, the converse is not true, not even for zero-dimensional Borel spaces). Homogeneity is a classical notion in topology, which has been studied in depth (see for example the survey [AvM1]). Here, we will focus on a much less studied notion.1 We will say that a space X is σ-homogeneous if there exist homogeneous subspaces Xn of X for n P ω such that X “ nPω Xn. Homogeneous spaces and countable spaces are trivial examples of σ-homogeneousŤ spaces. In [Os], Alexey Ostrovsky sketched a proof that every zero-dimensional Borel space is σ-homogeneous. Inspired by his result, we obtained the following theorem (see Theorem 8.2 for a stronger result), where AD denotes the Axiom of Determinacy. Theorem 1.1. Assume AD. Then every zero-dimensional space is σ-homogeneous. To complete the picture, it seemed natural to look for counterexamples under the Axiom of Choice, and for definable counterexamples under V “ L. This is exactly the content of the following two results (see Corollary 10.2 and Theorem 12.2), even though the analytic case still eludes us (see Question 13.2). Theorem 1.2. There exists a ZFC example of a zero-dimensional space that is not σ-homogeneous. Theorem 1.3. Assume V “ L. Then there exists a coanalytic zero-dimensional space that is not σ-homogeneous. While the proof of Theorem 1.2 is a rather straightforward diagonalization, the proof of Theorem 1.3 uses a method developed by the second-listed author in [Vi]. This method is a “black-box” version of the technique that is mostly known by the applications given by Miller in [Mi], and has spawned many more since then. It seems particularly fitting for this special issue to mention that the very first instance of this idea seems to have appeared in a paper by Erd˝os, Kunen and Mauldin (see [EKM, Theorems 13, 14 and 16]). Regarding Theorem 1.1, our fundamental tool will be Wadge theory, which was founded by William Wadge in his doctoral thesis [Wa1] (see also [Wa2]), and has become a classical topic in descriptive set theory. The application of Wadge theory to topology was pioneered by Fons van Engelen, based on the fine analysis of the Borel Wadge classes given by Alain Louveau in [Lo1]. In particular, in his remarkable doctoral thesis [vE2] (see also [vE1]), van Engelen gave a complete classification of the homogeneous zero-dimensional Borel spaces. Other articles in this vein are [vE3] and [Me3], where purely topological characterizations of Borel filters and Borel semifilters on ω are given.2 Notice that, as the analysis in [Lo1] is limited to the Borel context, the same limitation holds for all the results mentioned above. In fact, apart from the present article, the only applications of Wadge theory to topology that go beyond the

1 See however [vM], [AvM2] and [AvM3], where somewhat related questions are investigated. 2 There, by identifying each subset of ω with its characteristic function, ﬁlters and semiﬁlters on ω are viewed as subspaces of 2ω. ZERO-DIMENSIONAL σ-HOMOGENEOUSSPACES 3

Borel realm are given in [vEMS] and [CMM1]. In [vEMS], the authors take a very game-oriented approach (using Lipschitz-reduction and the machinery of ﬁlling in), while [CMM1] gives a more systematic structural analysis of the Wadge classes generated by zero-dimensional homogeneous spaces, based on ideas from Louveau’s unpublished book [Lo2]. Here, we will proceed in the spirit of the latter article. In fact, our main reference will be [CMM2], where the Wadge-theoretic portion of [Lo2] is generalized, and presented in full detail. Something that all topological applications of Wadge theory have in common (and this article is no exception) is that they ultimately rely on a theorem of Steel from [St1], which is discussed in Section 6. This theorem, loosely speaking, allows one to make the very considerable leap from Wadge equivalence to homeomorphism. We conclude this section by recalling two more topological notions that will appear naturally in the course of our investigations. A space X is rigid if |X|ě 2 and the only homeomorphism h : X ÝÑ X is the identity.3 See [vD, Appendix 2] and [MvMZ1] for an introduction to rigid spaces. A space X is strongly homogeneous (or h-homogeneous) if every non-empty clopen subspace of X is homeomorphic to X. This notion has been studied by several authors, both “instrumentally” and for its own sake (see the list of references in [Me1]). While it is well-known that every zero-dimensional strongly homogeneous space is homogeneous (see for example [vE2, 1.9.1] or [Me2, Proposition 3.32]), the other implication depends on set-theoretic assumptions (see [CMM1, Theorems 1.1 and 1.2]).

2. Preliminaries and notation Let Z be a set, and let Γ Ď PpZq. Define Γ “ tZzA : A P Γu. We will say that Γ is selfdual if Γ “ Γ. Also define ∆pΓq “ Γ X Γ. Given a function q f : Z ÝÑ W , A Ď Z and B Ď W , we will use the notation frAs“tfpxq : x P Au ´1 q q and f rBs“tx P Z : fpxq P Bu. We will use idX : X ÝÑ X to denote the identity function on a set X. Definition 2.1 (Wadge). Let Z be a space, and let A, B Ď Z. We will write A ď B if there exists a continuous function f : Z ÝÑ Z such that A “ f ´1rBs.4 In this case, we will say that A is Wadge-reducible to B. We will write A ă B if A ď B and B ę A. We will write A ” B if A ď B and B ď A. We will say that A is selfdual if A ” ZzA. Definition 2.2 (Wadge). Let Z be a space. Given A Ď Z, define AÓ“tB Ď Z : B ď Au.5 Given Γ Ď PpZq, we will say that Γ is a Wadge class if there exists A Ď Z such that Γ “ AÓ , and that Γ is continuously closed if AÓĎ Γ for every A P Γ. We will denote by NSDpZq the collection of all non-selfdual Wadge classes in Z.

3 The requirement that |X|ě 2 is not standard, but it will enable us to avoid trivialities, thus simplifying the statement of several results. 4 Wadge-reduction is usually denoted by ďW, which allows to distinguish it from other types of reduction (such as Lipschitz-reduction). Since we will not consider any other type of reduction in this article, we decided to simplify the notation. 5 We point out that AÓ is sometimes denoted by rAs (see for example [CMM1], [vE1], [vE2], [vE3] and [Lo1]). We decided to avoid this notation, as it conﬂicts with the notation for the Wadge degree of A, that is tB Ď Z : B ” Au. 4 ANDREAMEDINIANDZOLTAN´ VIDNYANSZKY´

Both of the above definitions depend of course on the space Z. Often, for the sake of clarity, we will specify what the ambient space is by saying, for example, that “A ď B in Z” or “Γ is a Wadge class in Z”. Notice that t∅u and tZu are trivial examples of Wadge classes in Z for every space Z. Our reference for descriptive set theory is [Ke]. In particular, we assume familiarity with the basic theory of Borel, analytic and coanalytic sets in Polish spaces, and mostly use the same notation as in [Ke]. For example, given a space Z, we use 0 0 0 Σ1pZq, Π1pZq, and ∆1pZq to denote the collection of all open, closed, and clopen subsets of Z respectively. We will denote by BorelpZq the collection of all Borel subsets of Z. Our reference for other set-theoretic notions is [Ku]. The classes defined below constitute the so-called difference hierarchy (or small Borel sets). For a detailed treatment, see [Ke, Section 22.E] or [vE2, Chapter 3]. Recall that, given spaces X and Z, a function j : X ÝÑ Z is an embedding if j : X ÝÑ jrXs is a homeomorphism.

Deﬁnition 2.3 (Kuratowski). Let Z be a space, let 1 ď η ă ω1 and 1 ď ξ ă ω1.

‚ Given a sequence of sets pAµ : µ ă ηq, deﬁne

tAµz ζăµ Aζ : µ ă η and µ is oddu if η is even, DηpAµ : µ ă ηq“ " ŤtAµz Ťζăµ Aζ : µ ă η and µ is evenu if η is odd. D 0Ť Ť 0 ‚ Deﬁne A P ηpΣξpZqq if there exist Aµ P ΣξpZq for µ ă η such that D 6 D 0 D 0 A “ ηpAµ : µ ă ηq. If Γ “ ηpΣξpZqq, we will denote Γ by ηpΣξpZqq. ‚ A space X is D pΣ0q if jrXsP D pΣ0pZqq for every space Z and embedding η ξ η ξ q q j : X ÝÑ Z.7 Next, we repeat some useful conventions from [vE2]. Given a topological property P, we will say that a space X is nowhere P if X is non-empty and every open subspace of X is P. For the sake of brevity, we will say that a space is complete if it is completely metrizable. We will say that a space X is strongly σ-complete if there exist complete closed subspaces Xn of X for n P ω such that X “ nPω Xn. D 0 We will only be interested in ηpΣξ q when η ď ω is even and ξ “Ť 2. The following characterization is a consequence of [vE2, Lemmas 3.1.3, 3.1.4 and 3.1.5] and their proofs. Recall that 2ω “ ω. Proposition 2.4 (van Engelen). Let X be a space, and let 1 ď k ď ω. Then the following conditions are equivalent: 0 ‚ X is D2kpΣ2q, ‚ X can be written as a countable union of k closed strongly σ-complete subspaces of X, ‚ There exist a compact space Z and an embedding j : X ÝÑ Z such that 0 jrXsP D2kpΣ2pZqq. Given a set A, we will denote by Aăω (respectively Aďω) the collection of all functions s : n ÝÑ A, where n ă ω (respectively n ď ω). Given s P Aăω, we will ω 8 use the notation Ns “tz P A : s Ď zu.

6 DZ Σ0 D Σ0 Our notation diﬀers slightly from van Engelen’s, as he uses η p ξ q instead of ηp ξ pZqq. 7 D Σ0 These spaces, for obvious reasons, are sometimes called absolutely ηp ξ q. 8 In all our applications, we will have A “ 2 or A “ ω. ZERO-DIMENSIONAL σ-HOMOGENEOUSSPACES 5

For an introduction to games, we refer the reader to [Ke, Section 20]. Here, we only want to give the precise definition of determinacy. Given a set A, a play of the game GpA, Xq is described by the diagram I a0 a2 ¨ ¨ ¨ II a1 a3 ¨ ¨ ¨ , ω in which an P A for every n P ω and X Ď A is called the payoff set. We will say that Player I wins this play of the game GpA, Xq if pa0,a1,...qP X. Player II wins if Player I does not win. A strategy for a player is a function σ : Aăω ÝÑ A. We will say that σ is a winning strategy for Player I if setting a2n “ σpa1,a3,...,a2n´1q for each n makes ω Player I win for every pa1,a3,...qP A . A winning strategy for Player II is defined similarly. We will say that the game GpA, Xq (or simply the set X) is determined if (exactly) one of the players has a winning strategy. For the purposes of this article, the case A “ ω is the only relevant one. Given Σ Ď Ppωωq, we will write DetpΣq to mean that every element of Σ is determined. The assumption DetpPpωωqq is known as the Axiom of Determinacy (briefly, AD). We will denote by BP the axiom stating that for every Polish space Z, every subset of Z has the Baire property. Using the arguments in [Ke, Section 21.C], it can be shown that AD implies BP. As usual, we will denote the Axiom of Choice by AC. It is well-known that AD is incompatible with AC (see for example [Ka, Propo- sition 28.1]). This is the reason why, throughout this article, we will be working in ZF ` DC.9 Recall that the principle of Dependent Choices (briefly, DC) states that if R is a binary relation on a non-empty set A such that for every a P A there exists ω b P A such that pb,aqP R, then there exists a sequence pa0,a1,...qP A such that pan`1,anqP R for every n P ω. An equivalent formulation of DC is that a relation ω R on a set A is well-founded iff there exists no sequence pa0,a1,...qP A such that pan`1,anqP R for every n P ω. Furthermore, DC implies the countable analogue of AC. To the reader who is unsettled by the lack of AC, we recommend [HR]. We conclude this section with some miscellaneous topological definitions and results. We will write X « Y to mean that the spaces X and Y are homeomorphic. A subset of a space is clopen if it is closed and open. A space is zero-dimensional if it is non-empty and it has a base consisting of clopen sets.10 A space X is a Borel space (respectively, analytic or coanalytic) if there exist a Polish space Z and an 1 embedding j : X ÝÑ Z such that jrXsP BorelpZq (respectively, jrXs P Σ1pZq or 1 jrXsP Π1pZq). It is well-known that a space X is Borel (respectively, analytic or 1 1 coanalytic) iff jrXs P BorelpZq (respectively, jrXs P Σ1pZq or jrXs P Π1pZq) for every Polish space Z and every embedding j : X ÝÑ Z (see [MZ, Proposition 4.2]). For example, by [Ke, Theorem 3.11], every Polish space is a Borel space. Given an infinite cardinal κ, we will say that a space X is κ-crowded if it is non-empty and |U|“ κ for every non-empty subset U of X. Furthermore, we will say that a subset X of a non-empty space Z is κ-dense in Z if |U X X| “ κ for every non-empty subset U of Z. The next proposition can be safely assumed to be folklore, while Lemma 2.6 follows easily from [Te, Theorem 2.4] (see also [Me1, Theorem 2 and Appendix A] or [Me2, Theorem 3.2 and Appendix B]).

9 The consistency of ZF ` DC ` AD can be obtained under suitable large cardinal assumptions (see [Ka, Proposition 11.13] and [Ne]). 10 The empty space has dimension ´1 (see [En, Section 7.1]). 6 ANDREAMEDINIANDZOLTAN´ VIDNYANSZKY´

Proposition 2.5. Assume BP. Let Z be a Polish space, and let X be a dense Baire subspace of Z. Then X is comeager in Z. Proof. Since X has the Baire property, we can write X “ GYM by [Ke, Proposition 0 8.23.ii], where G P Π2pZq and M is meager in Z. It will be enough to show that G is dense in Z. Assume, in order to get a contradiction, that there exists a non-empty open subset U of Z such that U X G “ ∅. Observe that U X X is a non-empty open subset of X because X is dense in Z. Furthermore, using the density of X again, it is easy to see that M “ M X X is meager in X. Since U X X Ď M, this contradicts the fact that X is a Baire space. 0 Lemma 2.6 (Terada). Let X be a space. Assume that X has a base B Ď ∆1pXq such that U « X for every U P B. Then X is strongly homogeneous.

3. The basics of Wadge theory The following two results (the first is commonly known as “Wadge’s Lemma”) are the most fundamental theorems of Wadge theory. For the proofs, see [CMM1, Lemma 3.2 and Theorem 3.3]. Lemma 3.1 (Wadge). Assume AD. Let Z be a zero-dimensional Polish space, and let A, B Ď Z. Then either A ď B or ZzB ď A. Theorem 3.2 (Martin, Monk). Assume AD. Let Z be a zero-dimensional Polish space. Then the relation ď on PpZq is well-founded. As a first application of Wadge’s Lemma, one can easily show that the largest antichains with respect to ď have size 2. These antichains are in the form tΓ, Γu for some Γ P NSDpZq, and they are known as non-selfdual pairs. Another application q is given by Lemma 3.3, whose simple proof is left to the reader. Proposition 3.4 gives the simplest non-trivial examples of Wadge classes.11 Lemma 3.3. Assume AD. Let Z be a zero-dimensional Polish space. If Γ Ď PpZq is non-selfdual and continuously closed then Γ P NSDpZq. Proposition 3.4. Let Z be an uncountable zero-dimensional Polish space, let 1 ď D 0 NSD η ă ω1 and 1 ď ξ ă ω1. Then ηpΣξpZqq P pZq. D 0 Proof. Set Γ “ ηpΣξ pZqq. Notice that Γ is continuously closed by [Ke, Exercise 22.26.i], while Γ is non-selfdual by by [Ke, Exercise 22.26.iii]. Therefore, the desired conclusion follows from Lemma 3.3. Observe that non-selfdual Wadge classes seem to suffer from the “pair of socks” problem. In less ridiculous words, given a non-selfdual Wadge class Γ, there is no obvious way to distinguish Γ from Γ. The following result gives an elegant solution to this problem (see [VW, Theorem 2] and [St2]). Recall that Γ Ď Ppωωq has the q separation property if whenever A and B are disjoint elements of Γ there exists C P ∆pΓq such that A Ď C Ď ωωzB. Theorem 3.5 (Steel, Van Wesep). Assume AD. For every Γ P NSDpωωq, exactly one of Γ and Γ has the separation property.

11 A more systematicq proof of Proposition 3.4 could be given by using Hausdorﬀ operations (see [CMM1, Sections 5-7]). ZERO-DIMENSIONAL σ-HOMOGENEOUSSPACES 7

The way in which we will apply the above theorem is rather amusing, as we will not care at all about what the separation property says speciﬁcally: we will only use the fact that it provides a canonical choice between Γ and Γ, thus allowing us to bypass AC (see the deﬁnition of HCpXq at the beginning of Section 8).12 q Next, we give the well-known analysis of the selfdual sets. Using Theorem 3.6 and Corollary 3.7, it is often possible to reduce the selfdual case to the non-selfdual case. For the proofs, see [CMM2, Theorem 5.4 and Corollary 5.5]. Theorem 3.6. Assume BP. Let Z be a zero-dimensional Polish space, let V P 0 0 ∆1pZq, and let A be a selfdual subset of Z. Assume that A R ∆1pZq. Then there 0 exist pairwise disjoint Vn P ∆1pV q for n P ω such that nPω Vn “ V and AXVn ă A in Z for each n. Ť Corollary 3.7. Assume AD. Let Z be a zero-dimensional Polish space, let V P 0 ∆1pZq, and let A be a selfdual subset of Z. Then there exist pairwise disjoint 0 Vn P ∆1pV q and non-selfdual An ă A in Z for n P ω such that nPω Vn “ V and nPωpAn X Vnq“ A X V . Ť Ť We conclude this section two closure properties. The elementary proof of Propo- sition 3.8 is left to the reader. The case Z “ ωω of Theorem 3.9 is due to Andretta, Hjorth and Neeman (see [AHN, Lemma 3.6.a]), however we refer to [CMM1, Lemma 12.3] for the proof of the more general version given here. Proposition 3.8. Let Z be a space, let Γ be a Wadge class in Z, and let A P Γ. 0 ‚ Assume that Γ ‰tZu. Then A X V P Γ for every V P ∆1pZq. 0 ‚ Assume that Γ ‰t∅u. Then A Y V P Γ for every V P ∆1pZq. Theorem 3.9. Assume AD. Let Z be an uncountable zero-dimensional Polish 0 space, and let Γ P NSDpZq. Assume that DnpΣ1pZqq Ď Γ whenever 1 ď n ă ω. 0 ‚ If A P Γ and C P Π1pZq then A X C P Γ. 0 ‚ If A P Γ and U P Σ1pZq then A Y U P Γ.

4. Relativization In this section we will set up the machinery of relativization, which will allow us to consider the “same” Wadge class in different ambient spaces. The results in this section are essentially due to Louveau and Saint-Raymond (see [LSR2, Section 4]), although a more systematic exposition was given in [CMM2, Sections 6 and 7].13 We refer to the latter for the proofs and a more extensive treatment. Definition 4.1 and Lemma 4.2 establish the foundations of this method, while Lemmas 4.3 and 4.4 give several “reassuring” and extremely useful properties of relativization. In particular, we will be often using Lemma 4.3 without mentioning it explicitly. For the proofs, see [CMM2, Section 6]. Definition 4.1 (Louveau, Saint-Raymond). Given a space Z and Γ Ď Ppωωq, define 1 ΓpZq“tA Ď Z : g´ rAsP Γ for every continuous g : ωω ÝÑ Zu.

12 Recall that we are working in ZF ` DC. 13 An alternative approach to relativization (although equivalent in practice) was given in [CMM1, Section 6], using Hausdorﬀ operations. The approach given here has the advantage of not having to rely on Van Wesep’s theorem (see [CMM1, Section 8]), which is far from elementary. 8 ANDREAMEDINIANDZOLTAN´ VIDNYANSZKY´

Lemma 4.2. Assume AD. Let Z be a zero-dimensional Polish space, and let Λ P NSDpZq. Then there exists a unique Γ P NSDpωωq such that ΓpZq“ Λ. Lemma 4.3. Let Γ Ď Ppωωq, and let Z and W be spaces. ‚ If f : Z ÝÑ W is continuous and B P ΓpW q then f ´1rBsP ΓpZq. ‚ If h : Z ÝÑ W is a homeomorphism then A P ΓpZq iff hrAsP ΓpW q. ‚ ΓpZq“ ΓpZq. ‚ If Γ is continuously closed then Γpωωq“ Γ. ~ q Lemma 4.4. Assume AD. Let Z and W be zero-dimensional Borel spaces such that W Ď Z, and let Γ P NSDpωωq. Then A P ΓpW q iff A X W “ A for some A P ΓpZq. r r Next, we state special cases of [CMM2, Theorems 7.1 and 7.2]. These results show that, when the ambient spaces Z and W are zero-dimensional, uncountable and Polish, the correspondence ΓpZq ÞÝÑ ΓpW q is an order-isomorphism between the non-selfdual Wadge classes in Z and the non-selfdual Wadge classes in W . Theorem 4.5. Assume AD. Let Z and W be zero-dimensional uncountable Borel spaces, and let Γ, Λ P NSDpωωq. Then ΓpZqĎ ΛpZq iff ΓpW qĎ ΛpW q. Theorem 4.6. Assume AD. Let Z be a zero-dimensional uncountable Polish space. Then NSDpZq“tΓpZq : Γ P NSDpωωqu.

5. The notion of level The aim of this section it to introduce an extremely useful tool in the analysis of non-selfdual Wadge classes. Essentially, this notion first appeared in [Lo1], but we will follow the approach of [LSR1], which was however limited to the Borel context (see also [Lo2, Section 7.3.4]). We begin with a preliminary definition.14 Definition 5.1 (Louveau, Saint-Raymond). Let Z be a space, let Γ Ď PpZq, and let ξ ă ω1. Define PUξpΓq to be the collection of all sets of the form

pAn X Vnq, nďPω 0 where each An P Γ, each Vn P ∆1`ξpZq, the Vn are pairwise disjoint, and nPω Vn “ Z. A set in this form is called a partitioned union of sets in Γ. Ť Deﬁnition 5.2 (Louveau, Saint-Raymond). Let Z be a space, let Γ Ď PpZq, and let ξ ă ω1. Deﬁne

‚ ℓpΓqě ξ if PUξpΓq“ Γ, ‚ ℓpΓq“ ξ if ℓpΓqě ξ and ℓpΓqğ ξ ` 1, ‚ ℓpΓq“ ω1 if ℓpΓqě η for every η ă ω1. We refer to ℓpΓq as the level of Γ. It is easy to see that ℓpΓqě 0 for every Wadge class Γ. Using [Ke, Theorem 22.4 and Exercise 37.3], one sees that the following hold for every uncountable Polish space Z:

14 ∆0 PU PU In [LSR1], the notation 1`ξ- is used instead of ξ, and λC is used instead of ℓ. ZERO-DIMENSIONAL σ-HOMOGENEOUSSPACES 9

‚ ℓpt∅uq “ ℓptZuq “ ω1, 0 0 ‚ ℓpΣ1`ξpZqq “ ℓpΠ1`ξpZqq “ ξ whenever ξ ă ω1, 1 1 ‚ ℓpΣnpZqq “ ℓpΠnpZqq “ ω1 whenever 1 ď n ă ω. We remark that the notion of level is better understood in the context of the fundamental Expansion Theorem (see [CMM2, Theorem 16.1]), and that it is possible (although far from easy) to show that for every non-selfdual Wadge Γ there exists ξ ď ω1 such that ℓpΓq“ ξ (see [CMM2, Corollary 17.2]). However, since we will not need these results, we will not say anything more about them. The following technical result will be needed in the proof of Lemma 8.1. It is simply a restatement of [CMM1, Lemma 14.3]. Lemma 5.3. Assume AD. Let Z be an uncountable zero-dimensional Polish space, let Γ P NSDpZq be such that ℓpΓq“ 0, and let X P Γ be codense in Z. Then there 0 exists a non-empty U P ∆1pZq and Λ P NSDpZq such that Λ Ĺ Γ and X X U P Λ.

6. Steel’s theorem and good Wadge classes As we mentioned in the introduction, the following theorem (which is a particular case of [St1, Theorem 2]) is one of our main tools. Given a Wadge class Γ in 2ω ω and X Ď 2 , we will say that X is everywhere properly Γ if X X Ns P ΓzΓ for every s P 2ăω. We will not give the definition of reasonably closed class, as it is an ad q hoc notion that would not be particularly enlightening. In fact, the only property of reasonably closed classes that we will need is the one given by Lemma 6.3. We refer the interested reader to [CMM1, Section 13]. Theorem 6.1 (Steel). Assume AD. Let Γ be a reasonably closed Wadge class in 2ω. Assume that X and Y are subsets of 2ω that satisfy the following conditions: ‚ X and Y are everywhere properly Γ, ‚ X and Y are either both meager in 2ω or both comeager in 2ω. Then there exists a homeomorphism h :2ω ÝÑ 2ω such that hrXs“ Y . The following notion first appeared as [CMM1, Definition 12.1], inspired by work of van Engelen. For a proof of Lemma 6.3, see [CMM1, Lemma 13.2]. Definition 6.2 (Carroy, Medini, Müller). Let Z be a space, and let Γ be a Wadge class in Z. We will say that Γ is good if the following conditions are satisfied: ‚ Γ is non-selfdual, 0 ‚ ∆pDωpΣ2pZqqq Ď Γ, ‚ ℓpΓqě 1. Lemma 6.3 (Carroy, Medini, Müller). Assume AD. Let Γ be a good Wadge class in 2ω. Then Γ is reasonably closed.

7. Zero-dimensional homogeneous spaces of low complexity In his thesis [vE2], van Engelen discovered that the behaviour of zero-dimensional homogeneous spaces changes drastically depending on whether their complexity is 0 15 higher or lower than ∆ “ ∆pDωpΣ2qq. While Wadge theory is the perfect tool for dealing with spaces of complexity above ∆, diﬀerent techniques are needed for

15 0 Notice that we have not quite defined what ∆pDωpΣ2qq means as a topological property (we will not need it in any of our statements or proofs). We hope that the reader will be satisfied with this vague indication. 10 ANDREAMEDINIANDZOLTAN´ VIDNYANSZKY´ spaces of complexity below ∆ (and these techniques work well in the more general 0 setting of ∆2 spaces). We will essentially follow the same subdivision into cases here, except that in 0 the proof of Lemma 8.1, it will be convenient to use DωpΣ2q as the dividing line (this is the reason why below we will consider Pω and Xω). We begin by defining a collection of relevant topological properties, taken from [vE2, Definition 3.1.7] (see also [vE2, Lemma 3.1.4]) and [vE2, Definition 3.1.8]. For convenience, we will 0 stipulate that D0pΣ2q is the property of being the empty space. Definition 7.1 (van Engelen). Given a space X and k P ω, we will say that: 0 ‚ X is P4k if X is the union of a D2kpΣ2q subspace and a complete subspace, 0 ‚ X is P4k`1 if X is D2pk`1qpΣ2q, P1 D 0 ‚ X is 4k`2 if X is the union of a 2kpΣ2q subspace, a complete subspace, and a countable subspace, P1 D 0 ‚ X is 4k`3 if X is the union of a 2pk`1qpΣ2q subspace and a countable subspace, P2 D 0 ‚ X is 4k`2 if X is the union of a 2kpΣ2q subspace, a complete subspace, and a σ-compact subspace, P2 D 0 ‚ X is 4k`3 if X is the union of a 2pk`1qpΣ2q subspace and a σ-compact subspace. It will also be useful to define the following: P1 ‚ X is ´2 if X has size at most 1, P1 ‚ X is ´1 if X is countable, P2 ‚ X is ´2 if X is compact, P2 ‚ X is ´1 if X is σ-compact, 0 ‚ X is Pω if X is DωpΣ2q. To indicate one of these properties generically (that is, in case we do not know piq whether the superscript i Pt1, 2u is present or not) we will use the notation Pn . Declare a linear order ă on these properties as follows: P1 ă P1 ă P2 ă P2 ă ´2 ´1 ´2 ´1 ¨ ¨ ¨ ă P ă P ă P1 ă P1 ă P2 ă P2 ă ă P ¨ ¨ ¨ 4k 4k`1 4k`2 4k`3 4k`2 4k`3 ¨ ¨ ¨ ω. piq For every n P t´2, ´1uY ω Ytωu and i Pt1, 2u such that Pn is defined, we will piq consider a class of spaces Xn , and use the same notational convention. Instead of giving their definitions, we will use the characterization given by the following theorem (see [vE2, Theorem 3.4.24] for the case n ă ω, and [vE2, Definition 3.5.7] for the case n “ ω). In fact, this characterization is at the same time more understandable and more suitable for the applications given here. Theorem 7.2 (van Engelen). Let n P t´2, ´1uY ω Ytωu and i Pt1, 2u. Then, for a zero-dimensional space X, the following conditions are equivalent: piq ‚ X P Xn , piq pjq ‚ X is Pn and nowhere Pm for every m P t´2, ´1uYω Ytωu and j Pt1, 2u pjq piq such that Pm ă Pn . piq The fundamental property of the classes Xn that we will need is given by the following result (see [vE2, Theorem 3.4.13] for the case n ă ω and [vE2, Theorem 3.5.9] for the case n “ ω). ZERO-DIMENSIONAL σ-HOMOGENEOUS SPACES 11

Theorem 7.3 (van Engelen). Let n P t´2, ´1uY ω Ytωu and i Pt1, 2u. Then, up piq to homeomorphism, the class Xn contains exactly one element, which is strongly homogeneous.

piq A few clarifications are in order. First of all, van Engelen did not define Pn or piq Xn for n “ ´2. In particular, he did not include them in his statements of the above two theorems. However, it is not hard to realize that they naturally fit into the context described here. In fact, using the classical characterizations of 2ω, Q and Q ˆ 2ω (see [vE2, Theorems 2.1.1, 2.4.1 and 2.4.5]), one sees that:

X 1 ‚ ´2 is the class of spaces of size 1, X 1 ‚ ´1 is the class of spaces that are homeomorphic to Q, X 2 ω ‚ ´2 is the class of spaces that are homeomorphic to 2 , X 2 ω ‚ ´1 is the class of spaces that are homeomorphic to Q ˆ 2 .

Furthermore, the same characterizations show that Theorem 7.3 holds for these 2 classes as well. Finally, we remark that the class Xω is denoted Xω by van Engelen, piq but here we prefer to make sure that the notation for each class Xn always matches piq the one for the corresponding property Pn . The following diagram (which is taken from [vE2, page 28]) illustrates the ﬁrst piq few classes Xn . For a concrete description of the spaces T and S (introduced by van Douwen and van Mill respectively), see [Me3, Section 5].

X 1 ω X 2 1 P ´2 2 P ´2

X 1 ω X 2 Q P ´1 Q ˆ 2 P ´1

ω ω P X0

ω Q ˆ ω P X1

1 2 T P X2 S P X2

1 2 Q ˆ T P X3 Q ˆ S P X3

. . 12 ANDREAMEDINIANDZOLTAN´ VIDNYANSZKY´

8. The positive result In this section, we will put together all the tools accumulated so far to obtain the positive result announced in the introduction (namely, Theorem 8.2). We begin by deﬁning, given a subspace X of 2ω, a subspace HCpXq of X which is Homogeneous (by Lemma 8.1) and Clopen (by construction). Throughout this section, we will 0 ω ω assume that an enumeration ∆1p2 q“tCk : k P ωu of the clopen subsets of 2 has been ﬁxed. We remark that HCpXq is not canonical, as it will depend on this enumeration. Case 1: X “ ∅. In this case, simply set HCpXq“ ∅. 0 Case 2: X has a non-empty open DωpΣ2q subspace. 16 piq Let P be the ă-minimal property Pn , where n P t´2, ´1uY ω Ytωu and 0 ω i Pt1, 2u, such that there exists C P ∆1p2 q satisfying the following conditions: ‚ X X C ‰ ∅, ‚ X X C is P.

Now fix C “ Ck, where k P ω is minimal such that the two conditions above are satisfied. Finally, define HCpXq“ X X C. 0 Case 3: X is nowhere DωpΣ2q. ω 0 ω Fix a Ď-minimal Γ P NSDpω q such that there exists C P ∆1p2 q satisfying the following conditions: ‚ X X C ‰ ∅, ‚ X X C P Γp2ωq. In case there are two possible choices for Γ (which would have to be a non-selfdual pair by Lemma 3.1), pick the one that has the separation property (recall Theorem 3.5). Now fix C “ Ck, where k P ω is minimal such that the two conditions above plus the following one are satisfied: ‚ Either X X C is a Baire space or X X C is a meager space. Finally, define HCpXq“ X X C. Lemma 8.1. Assume AD. Let X be a subspace of 2ω. Then HCpXq is a strongly homogeneous clopen subspace of X. Proof. Set U “ HCpXq. The fact that U is clopen in X is clear. To see that U is strongly homogeneous, we will consider the three cases that appear in the definition of HCpXq. Case 1: X “ ∅. The desired conclusion holds trivially in this case. 0 Case 2: X has a non-empty open DωpΣ2q subspace. Let P be as in the definition of HCpXq. Using the minimality of P and the fact piq that each Pn is inherited by clopen subspaces, it is straightforward to check that pjq U is P and nowhere Pm for every m P t´2, ´1uY ω Ytωu and j Pt1, 2u such that pjq Pm ă P. By Theorems 7.2 and 7.3, it follows that U is strongly homogeneous. 0 Case 3: X is nowhere DωpΣ2q.

16 The ordering ă is described in Section 7. ZERO-DIMENSIONAL σ-HOMOGENEOUS SPACES 13

Let C and Γ be as in the definition of HCpXq. Throughout the rest of this proof, we will write cl to denote closure in 2ω. Let K “ clpUq, and observe that K « 2ω. ω 0 ω Claim 1. If Λ P NSDpω q and there exists V P ∆1p2 q such that X X V ‰ ∅ and ω 0 ω ω X X V P Λp2 q, then DωpΣ2p2 qq Ď Λp2 q. 0 ω Let Λ and V be as in the statement of the claim. Notice that XXV R DωpΣ2p2 qq q 0 0 ω by Proposition 2.4, since X is nowhere DωpΣ2q. Since DωpΣ2p2 qq is a Wadge class by Proposition 3.4, the desired result follows from Lemma 3.1. ω 0 ω Claim 2. U X V is non-selfdual in 2 for every V P ∆1p2 q. 0 ω Pick V P ∆1p2 q. The claim is trivial if U X V “ ∅, so assume that U X V ‰ ∅. Assume, in order to get a contradiction, that U XV is selfdual. By Corollary 3.7, we 0 ω can fix pairwise disjoint Vn P ∆1p2 q and non-selfdual An ă U X V for n P ω such ω that nPω Vn “ 2 and nPωpAn X Vnq “ U X V . Fix n such that An X Vn ‰ ∅. 0 NoticeŤ that An ‰ Z becauseŤ X is nowhere DωpΣ2q, hence An XVn ď An ă U XV P ω Γp2 q by Proposition 3.8. Since An X Vn “ U X V X Vn “ X X C X V X Vn, this contradicts the minimality of Γ. Claim 3. Γp2ωq is a good Wadge class. The fact that Γp2ωq in non-selfdual follows from Theorem 4.6, as Γ is non-selfdual 0 ω ω by construction. Furthermore, one sees from Claim 1 that ∆pDωpΣ2p2 qqq Ď Γp2 q. It remains to show that ℓpΓp2ωqq ě 1. Since K « 2ω, it will be enough to show that ℓpΓpKqq ě 1. So assume, in order to get a contradiction, that ℓpΓpKqq “ 0. Notice 0 that U is codense in K because X is nowhere DωpΣ2q. Therefore, it is possible to apply Lemma 5.3. Together with Lemma 4.2 and Theorem 4.5, this yields a 0 ω non-empty V P ∆1pKq and Λ P NSDpω q such that Λ Ĺ Γ and U X V P ΛpKq. Set W “ U X V , and notice that W ‰ ∅ because U is dense in K. By Lemma 4.4, there exists W P Λp2ωq such that W X K “ W . Furthermore, we must have 0 0 Σ2pKqĎ ΛpKq, otherwise we would have W P Π2pKq by Lemma 3.1, contradicting Ă 0 Ă ω 0 ω ω the fact that X is nowhere DωpΣ2q. Since K « 2 , it follows that Σ2p2 qĎ Λp2 q. In particular, it is possible to apply Theorem 3.9, which yields W “ W XK P Λp2ωq. Finally, using the compactness of K, it is easy to see that W is in the form X X D 0 ω Ă for some D P ∆1p2 q, which contradicts the minimality of Γ. Finally, thanks to Claim 3 and Lemma 6.3, we are in a position to apply Theorem 6.1 to obtain the strong homogeneity of U. By Lemma 2.6, it will be enough to 0 ω show that D X U « U for every D P ∆1p2 q such that D X U ‰ ∅. So fix such a D, set L “ clpD X Uq, and observe that L « 2ω. Recall that U is either a Baire space or a meager space by construction. In the first case, Proposition 2.5 shows that U is a comeager subset of K, hence D X U is a comeager subset of L. In the second case, it is clear that U is a meager subset of K, and that D X U is a meager subset of L. Therefore, the next two claims will conclude the proof. Claim 4. U is everywhere properly ΓpKq. 0 ω Let V P ∆1p2 q be such that U X V ‰ ∅, and set W “ U X V . Notice that W P Γp2ωq by Proposition 3.8, hence W P ΓpKq by Lemma 4.4. Now assume, in order to get a contradiction, that W P ΓpKq. By Lemma 4.4, we can fix W P Γp2ωq such that W X K “ W . By Claim 1, it possible to apply Theorem 3.9, which yields q Ă q W “ W X K P Γp2ωq. Since W is non-selfdual in 2ω by Claim 2, this contradicts Ă the minimality of Γ. Ă q Claim 5. D X U is everywhere properly ΓpLq. The proof of this claim is similar to the proof of Claim 4. 14 ANDREAMEDINIANDZOLTAN´ VIDNYANSZKY´

Theorem 8.2. Assume AD. Then every zero-dimensional space is σ-homogeneous. More precisely, every zero-dimensional space X can be written as a countable disjoint union of closed strongly homogeneous subspaces of X. Proof. Let X be a zero-dimensional space. Without loss of generality, assume that ω X is a subspace of 2 . Using transfinite recursion, define Xα for every ordinal α as follows: ‚ X0 “ X, ‚ Xα`1 “ XαzHCpXαq, ‚ Xγ “ αăγ Xα, if γ is a limit ordinal. Since the Xα formŞ a decreasing sequence of closed subsets of X, we can fix δ ă ω1 such that Xα “ Xδ for every α ě δ. If we had Xδ ‰ ∅, then we would have Xδ`1 Ĺ Xδ by the definition of HCpXδq. Hence Xδ “ ∅, which implies

X “ HCpXαq. αďăδ

Since HCpXαq is strongly homogeneous for every α by Lemma 8.1, the proof is concluded. We conclude this section by pointing out that [vEMS, Theorem 2.4] follows very easily from Lemma 8.1. Theorem 8.3 (van Engelen, Miller, Steel). Assume AD. Then there exist no zero- dimensional rigid space. Proof. Let X be a zero-dimensional space such that |X|ě 2, and assume without loss of generality that X is a subspace of 2ω. Set U “ HCpXq and V “ HCpXzUq, and observe that U and V are strongly homogeneous clopen subspaces of X by Lemma 8.1. It is clear that X is not rigid if |U| ě 2 or |V | ě 2, so assume that |U|ď 1 and |V |ď 1. Since |X|ě 2, it follows that |U| “ |V |“ 1. Therefore X has at least two isolated points, which also implies that X is not rigid.

9. Hereditarily rigid spaces In this section, we introduce two strengthenings of the standard notion of rigidity that will appear naturally in the remainder of this article. Usually, given a topological property P, the hereditarily P spaces are those whose all subspaces satisfy P. In the case P “ rigidity, following this approach too strictly only leads to trivialities. However, as in the following definition, a small tweak is sufficient to yield two interesting properties. Definition 9.1. A space X is c-hereditarily rigid if X is c-crowded and every c- crowded subspace of X is rigid. A space X is strongly c-hereditarily rigid if X is c-crowded and the only homeomorphisms between c-crowded subspaces of X are of the form idS for some c-crowded subspace S of X. It is clear that every c-hereditarily rigid space is rigid, and that every strongly c-hereditarily rigid space is c-hereditarily rigid (see also Corollary 9.5 and Question 13.4). The existence in ZFC of a strongly c-hereditarily rigid space is the content of the next section. Proposition 9.3 is the reason why the above notions are relevant in our context. It will easily follow from Proposition 9.2, which can be safely assumed to be folklore. ZERO-DIMENSIONAL σ-HOMOGENEOUS SPACES 15

Proposition 9.2. Let κ be an uncountable cardinal. Then every homogeneous space of size κ is κ-crowded. Proof. Let X be a homogeneous space such that |X|“ κ. Assume, in order to get a contradiction, that U is a non-empty open subset of X such that |U|ă κ. Set U “thrUs : h is a homeomorphism of Xu, and observe that U is a cover of X by homogeneity. By considering a countable subcover of U, one sees that |X| ď |U| ¨ ω ă κ, which is a contradiction. Proposition 9.3. Let X be a c-hereditarily rigid space. Then X is not σ-homogeneous.

Proof. Assume, in order to get a contradiction, that X “ nPω Xn, where each Xn is homogeneous. Fix n such that |Xn| “ c, and observeŤ that Xn is c-crowded by Proposition 9.2. It follows that Xn is rigid, which is a contradiction. We conclude this section by investigating these notions a little further, although these observations will not be needed later on. Recall that a subset X of 2ω is Bernstein if K X X ‰ ∅ and K X p2ωzXq ‰ ∅ for every perfect subset K of 2ω. The following result first appeared as [MvMZ1, Theorem 6] (see also the proof of [MvMZ1, Theorem 5] for the fact that X is Bernstein). Theorem 9.4 (Medini, van Mill, Zdomskyy). There exists a ZFC example of a subspace X of 2ω with the following properties, where Y “ 2ωzX: ‚ X is Bernstein, ‚ X is rigid, ‚ Y is homogeneous. Corollary 9.5. There exists a ZFC example of a zero-dimensional c-crowded rigid space that is not c-hereditarily rigid. Proof. Let X be the subspace of 2ω given by Theorem 9.4, and set Y “ 2ωzX. Using the homogeneity of Y , one can fix a homeomorphism h : Y ÝÑ Y that is not 0 ω the identity. By [Ke, Exercise 3.10], we can fix G P Π2p2 q and a homeomorphism h : G ÝÑ G such that h Ď h. Notice that X and Y are both c-dense in 2ω because they are Bernstein. It r r follows that G is comeager in 2ω, hence U X G contains a copy of 2ω for every non-empty open subset U of 2ω. Since X is Bernstein, this shows that G X X is dense in 2ω. At this point, it is easy to realize that h æ G X X is a homeomorphism of G X X other than the identity. r We remark that, assuming V “ L, one can even obtain an analytic space as in the above corollary (see Theorem 12.4). However, we do not know whether V “ L yields a coanalytic version of this counterexample (see Question 13.5).

10. A counterexample in ZFC In this section, we will construct in ZFC a space with the strongest of the rigidity properties that we previously deﬁned. Corollary 10.2 shows how this is related to main topic of this article. The proof of Theorem 10.1 was inspired by [MvMZ2, Proposition 2.10], which shows that the space of ω1 Cohen reals is rigid. While this proof was discovered with the help of elementary submodels, we subsequently realized that their use can be comfortably avoided. 16 ANDREAMEDINIANDZOLTAN´ VIDNYANSZKY´

Theorem 10.1. There exists a ZFC example of a zero-dimensional strongly c- hereditarily rigid space. Proof. By passing to a c-crowded subspace, it will be enough to construct a subspace X of 2ω such that |X| “ c and the only homeomorphisms between c-crowded subspaces of X are of the form idS for some c-crowded subspace S of X. Fix an enumeration thα : α ă cu of all homeomorphisms hα : Gα ÝÑ Hα, where 0 ω ω ω Gα,Hα P Π2p2 q. Also assume that h0 :2 ÝÑ 2 is the identity. Using transﬁnite ω recursion, pick xα P 2 for α ă c such that the following condition are satisﬁed:

(1) xα ‰ hβpxγ q for every β,γ ă α such that xγ P Gβ, ´1 (2) xα ‰ hβ pxγ q for every β,γ ă α such that xγ P Hβ. In the end, set X “txα : α ă cu. Notice that xα ‰ xβ whenever α ‰ β by our choice of h0. In particular, we see that |X|“ c. Now fix c-crowded subspaces S and T of X, and let h : S ÝÑ T be a homeomorphism. By [Ke, Theorem 3.9], we can fix δ ă c such that h Ď hδ. We will show that hpxαq“ xα for every α ą δ such that xα P S. Since S is c-crowded, this will imply that S “ T and h “ idS, concluding the proof. So fix α ą δ such that xα P S, and let β ă c be such that hpxαq“ xβ. We will prove that β “ α by showing that every other case leads to a contradiction. Case 1: β ą α. This would violate condition p1q in the construction of xβ. Case 2: β ă α. ´1 Since xα “ hδ pxβ q, this would violate condition p2q in the construction of xα.

Corollary 10.2. There exists a ZFC example of a zero-dimensional space X that is not σ-homogeneous. Proof. This follows from Theorem 10.1 and Proposition 9.3.

11. Preliminaries on V “ L We will assume some familiarity with the basic theory of L (see [Ku]) and basic recursion theory (see [Od]). We will say that S Ď ωω is cofinal in the Turing degrees if for every x P ωω there exists y P S such that x is recursive in y. Given x P ωω, we x will denote by ω1 the least ordinal that is not recursive in x. Furthermore, we will ω x say that x P ω is self-constructible if x P Lpω1 q. Given X Ď Z ˆ W and z P Z, we will use the notation Xz “tw P W : pz, wqP Xu for the vertical section of X at z. Theorem 11.1 is essentially a restatement of [Vi, Theorem 1.3]. The only significant difference is that we added that X consists of self-constructible reals, but this is clear from the proof (thanks to [Vi, Lemma 3.2]). While this fact is described in [Vi, page 173] as “one of the weaknesses of the method”, here it will be a crucial ingredient in our arguments (see the proof of Lemma 12.1). Following [Vi, Definition 1.2], given F Ď M ďω ˆ B ˆ M, where M and B are sets of size ω1, we will say that X Ď M is compatible with F if there exist enumerations B “ tpα : α ă ω1u, X “ txα : α ă ω1u and, for every α ă ω1, a ďω sequence Aα P M that is an enumeration of txβ : β ă αu in type ď ω such that xα P FpAα,pαq for every α ă ω1. Intuitively, one should think of Aα as enumerating the portion of the desired set X constructed before stage α. The section FpAα,pαq consists of the admissible candidates to be added at stage α, where pα encodes the current condition to be satisfied. ZERO-DIMENSIONAL σ-HOMOGENEOUS SPACES 17

Theorem 11.1 (Vidnyánszky). Assume V “ L. Let M “ ωω, and let B be an uncountable Borel space.17 Assume that F Ď M ďω ˆ B ˆ M is coanalytic, and ďω that for all pA, pq P M ˆ B the section FpA,pq is cofinal in the Turing degrees. Then there exists a coanalytic X Ď M consisting of self-constructible reals that is compatible with F . Lemma 11.3 already gives our desired counterexample, although the verification will be carried out in the next section. We will need the following fact, which is an immediate consequence of [MW, Example 4.27]. ω ω x y Lemma 11.2. The relation tpx, yqP ω ˆ ω : ω1 ă ω1 u is coanalytic. Lemma 11.3. Assume V “ L. Then there exists X Ď ωω such that: ‚ X is coanalytic, ‚ X is c-dense in ωω, ‚ Every element of X is self-constructible, x y ‚ If x, y P X and x ‰ y then ω1 ‰ ω1 . Proof. Our plan is to apply Theorem 11.1 with B “ ωăω ˆ 2ω, where ωăω has the discrete topology. The purpose of B is simply to ensure that X will be c-dense in M “ ωω. More precisely, the factor ωăω will allow us to specify a basic clopen set, while the factor 2ω will guarantee that every basic clopen set is met c-many times. Let π : B ÝÑ ωăω denote the projection on the first coordinate, and set R “ ω ω x y ďω tpx, yqP ω ˆ ω : ω1 ă ω1 u. Define F Ď M ˆ B ˆ M by declaring

pA, p, xqP F iff x P Nπppq and @n P ω pApnq R xq . Using Lemma 11.2, it is straightforward` to check that F is coanalytic.˘ To check that ďω each FpA,pq is cofinal in the Turing degrees, pick pA, pqP M ˆB and z P M. First, 1 1 Apnq x 2 1 fix x P M such that ω1 ă ω1 for each n, then let x P M code x on the even coordinates and z on the odd coordinates. Finally, set x “ s"x2 æ pωznq, where s “ πppq : n ÝÑ ω. Since finite modifications do not affect Turing-reducibility, it is clear that x P Ns is as desired.

12. Definable counterexamples under V “ L In this section, we will show that the set/recursion-theoretic properties of the space given by Lemma 11.3 have significant topological consequences. The definable counterexample that we promised in the introduction will immediately follow (see Theorem 12.2). As the reader could certainly have guessed, we will say that a space X is σ-homogeneous with Borel witnesses if there exist homogeneous Xn P BorelpXq for n P ω such that X “ nPω Xn. Similarly, one can define what σ-homogeneous with closed witnesses means.Ť Lemma 12.1. Assume V “ L. Let X Ď ωω be such that the following conditions are satisfied: (1) X is coanalytic, (2) X is c-dense in ωω, (3) Every element of X is self-constructible, x y (4) If x, y P X and x ‰ y then ω1 ‰ ω1 .

17 More generally, one could replace ωω with any other uncountable Polish space with a natural notion of Turing reducibility. 18 ANDREAMEDINIANDZOLTAN´ VIDNYANSZKY´

Set Y “ ωωzX. Then: ‚ X and Y are c-crowded, ‚ X is strongly c-hereditarily rigid, ‚ X is not σ-homogeneous, ‚ Y is rigid but not c-hereditarily rigid, ‚ Y is not σ-homogeneous with Borel witnesses.

Proof. The fact that X is c-crowded follows trivially from condition p2q. Next, we will show that the following condition holds: f If h : S ÝÑ T is a homeomorphism without fixed points, where S,T Ď ωω, then tx P S X X : hpxqP Xu is countable. 0 ω Let h be as above. By [Ke, Theorem 3.9], we can fix G, H P Π2pω q and a home- 1 omorphism h : G ÝÑ H such that h Ď h. Also fix δ ă ω1 such that h and h´ are coded in Lpδq. Assume, in order to get a contradiction, that hpxq P X for r r r r uncountably many x P S X X. In particular, by condition p4q, we can fix x P S X X x x such that hpxq P X and ω1 ě δ. Set y “ hpxq. Since x P Lpω1 q by condition p3q, x y x it follows that y “ hpxq P Lpω1 q, hence ω1 ď ω1 . On the other hand, the same argument applied to h´1 shows that ωx ď ωy. Therefore ωx “ ωy, which implies r 1 1 1 1 x “ y by condition p4q. This contradicts the assumption that h has no fixed points, r showing that condition f holds. In order to prove that X is strongly c-hereditarily rigid, fix c-crowded subspaces S and T of X, and let h : S ÝÑ T be a homeomorphism. If there existed x P S such that hpxq ‰ x, then condition f would be easily contradicted. Therefore S “ T and h “ idS. The fact that X is not σ-homogeneous now follows from Proposition 9.3. Observe that, since X is c-hereditarily rigid, it cannot contain copies of 2ω. By condition p1q plus standard arguments involving the property of Baire (see [Ke, Theorem 21.6 and Proposition 8.26]), it follows that X is meager in ωω. Therefore Y is comeager in ωω, hence c-crowded. Since analytic sets have the perfect set property (see [Ke, Theorem 29.1]), one sees that Y is not c-hereditarily rigid. Next, we will show that Y is rigid. Let h : Y ÝÑ Y be a homeomorphism. By 0 ω [Ke, Exercise 3.10], we can fix G P Π2pω q and a homeomorphism h : G ÝÑ G such that h Ď h. It will be enough to show that G X X is c-dense in ωω. In fact, this r will imply that h æ pG X Xq is the identity by the c-hereditary rigidity of X, thus r showing that h is the identity. So pick a non-empty open subset U of ωω. Notice r that U X X “ pU X X X Gq Y pUzGq has size c by condition p2q. Therefore, either r U X X X G has size c or UzG has size c. Since UzG P Borelpωωq and X cannot contain copies of 2ω, one sees that U X X X G must have size c, as desired. It remains to show that Y is not σ-homogeneous with Borel witnesses. Assume, Borel in order to get a contradiction, that Y “ nPω Yn, where each Yn P pXq is 0 ω ω homogeneous. Fix a countable base B Ď ∆1Ťpω q for ω . Denote by I the collection of all pU,nqP B ˆ ω such that there exists h with the following properties: 0 ω ‚ h : G ÝÑ H is a homeomorphism, where G, H P Π2pω q, ‚ h has no fixed points, ‚ G X Yn “ U X Yn, ‚ hrG X Yns“ H X Yn. ZERO-DIMENSIONAL σ-HOMOGENEOUS SPACES 19

For every i P I, fix hi : Gi ÝÑ Hi as above. Given i P I, we will denote by ni the unique element of ω such that i “ pU,niq for some U P B. Define 1 ω X “ tx P ω : x R Gi or px P Gi and hipxqP Y zYni qu, čiPI 1 1 and observe that X is analytic because each Yn P BorelpY q. We claim that X ∆X is countable. This will conclude the proof, because X is not analytic. If x P Yn for some Yn of size at least 2 then, using the homogeneity of Yn and [Ke, Theorem 3.9], it is easy to see that there exists i P I such that ni “ n and x P Gi. This shows that X1zX is countable. On the other hand, given i P I, using condition f, it is easy to realize that hipxq P Y zYni for all but countably many x P Gi X X. Since i was arbitrary, this shows that XzX1 is countable. Obviously, the previous paragraph also shows that Y is not σ-homogeneous with closed witnesses. We conclude by giving an alternative proof of this fact. Assume, in order to get a contradiction, that Y “ nPω Yn, where each Yn is a homogeneous ω closed subspace of Y . Since Y is non-meagerŤ in ω , we can fix n P ω such that Yn has non-empty interior in Y . This means that there exists a non-empty U Ď Yn that is both clopen in Y and clopen in Yn. Using the fact that Yn is homogeneous and zero-dimensional, it is easy to see that U is homogeneous. Since U is clopen in Y and it contains more than one point, this contradicts the fact that Y is rigid. Combining Lemmas 11.3 and 12.1 immediately yields the following three results. These results respectively concern σ-homogeneity, classical rigidity, and the notions of hereditary rigidity introduced in Section 9. Notice that Theorem 12.4 can be viewed as a sharper version of Theorem 12.3. The latter first appeared as [vEMS, Theorem 2.6]. Theorem 12.2. Assume V “ L. Then there exists a zero-dimensional coanalytic space that is not σ-homogeneous. Furthermore, there exists a zero-dimensional analytic space that is not σ-homogeneous with Borel witnesses. Theorem 12.3 (van Engelen, Miller, Steel). Assume V “ L. Then there there exist both analytic and coanalytic examples of a zero-dimensional rigid space. Theorem 12.4. Assume V “ L. Then there exists a zero-dimensional coanalytic space that is strongly c-hereditarily rigid. Furthermore, there exists a zero- dimensional analytic space that is c-crowded and rigid, but not c-hereditarily rigid.

13. Final remarks and open questions We begin by remarking that the results of Section 8 are “local” in nature. To express this precisely, recall from [CMM2, Definition 3.1] that a nice topological pointclass is a function18 Σ that satisfies the following requirements: ‚ The domain of Σ is the class of all spaces, ‚ ΣpZqĎ PpZq for every space Z, ‚ ΣpZq is closed under complements and finite unions for every space Z, ‚ BorelpZqĎ ΣpZq for every space Z, ‚ If f : Z ÝÑ W is a Borel function and B P ΣpW q then f ´1rBsP ΣpZq,

18 Here, the term “function” is an abuse of terminology, as each nice topological pointclass is a proper class. Therefore, any theorem that mentions them is strictly speaking an inﬁnite scheme. Nice topological pointclasses are simply a convenient expositional tool that allows one to simultaneously state the Borel, Projective, and full-Determinacy versions of a theorem. 20 ANDREAMEDINIANDZOLTAN´ VIDNYANSZKY´

‚ For every space Z, if jrZsP ΣpW q for some Borel space W and embedding j : Z ÝÑ W , then jrZs P ΣpW q for every Borel space W and embedding j : Z ÝÑ W . The following are the most important examples of nice topological pointclasses (this can be veriﬁed using [Ke, Exercise 37.3] and the methods of [MZ, Section 4]): ‚ ΣpZq“ BorelpZq for every space Z, 1 19 ‚ ΣpZq“ 1ďnăω ΣnpZq for every space Z, ‚ ΣpZq“ ŤPpZq for every space Z. Since all the relevant results from [CMM2] are formulated as below using nice topological pointclasses, and similar formulations could be given of Theorems 3.5 and 6.1, as well as of Lemmas 5.3 and 6.3, it is straightforward to check that the arguments of Section 8 actually yield the following result. Theorem 13.1. Let Σ be a nice topological pointclass, and assume that DetpΣpωωqq holds. Then: ‚ Every X P Σp2ωq is σ-homogeneous, ‚ No X P Σp2ωq of is rigid. Notice that Theorems 8.2 and 8.3 can be obtained by setting Σ “ P in the above result, while [Os, Theorem 1] and [vE2, Theorem 5.1] can be obtained by setting Σ “ Borel. We conclude the article with several open questions. The following seems to be the most pressing one, as it is the only obstacle to having a complete picture of σ-homogeneity in the realm of zero-dimensional spaces.20 We conjecture that the answer is “yes”, with analytic witnesses, and that this could be shown using ideas from the proof of Theorem 6.1. Question 13.2. Is every zero-dimensional analytic space σ-homogeneous? One of the themes of this article is that, in the context of zero-dimensional spaces, there seems to be a close parallel between rigidity and the lack of σ-homogeneity (for example, under AD, these notions are vacuously equivalent). Therefore, the following question seems natural. Question 13.3. Is there a ZFC example of a zero-dimensional σ-homogeneous rigid space? At least under additional set-theoretic assumptions? While Corollary 9.5 shows that rigidity is strictly weaker than c-hereditary rigidity in ZFC, we do not know whether the latter notion can be distinguished from its strong version. Furthermore, we do not know whether it is possible to obtain a coanalytic version of the space given by this corollary. The following questions ask for these counterexamples. Notice that, in order to answer Question 13.5, one could try to give a coanalytic version of the rigid space given by Theorem 9.4. Question 13.4. Is there a ZFC example of a zero-dimensional c-hereditarily rigid space that is not strongly c-hereditarily rigid? At least under additional set- theoretic assumptions?

19 Σ1 Following [Ke, page 315], we will say that X P npZq if there exist a Polish space W , an Σ1 embedding j : Z ÝÑ W and X P npW q such that jrXs“ X X jrZs. 20 Notice however that we do have a complete picture in the case of σ-homogeneity with closed r r witnesses. ZERO-DIMENSIONAL σ-HOMOGENEOUS SPACES 21

Question 13.5. Assuming V “ L, is there a zero-dimensional coanalytic rigid space that is not c-hereditarily rigid? Since Theorem 8.2 gives a seemingly stronger property than mere σ-homogeneity, one is left wondering whether the stronger versions can be distinguished from the standard one. The following two questions ask for such counterexamples. As the reader could certainly have guessed, we will say that a space X is σ-homogeneous with pairwise disjoint witnesses if there exist homogeneous pairwise disjoint subspaces Xn of X for n P ω such that X “ nPω Xn. Question 13.6. Is there a ZFC exampleŤ of a zero-dimensional σ-homogeneous space that is not σ-homogeneous with closed witnesses? At least under additional set-theoretic assumptions? Question 13.7. Is there a ZFC example of a zero-dimensional σ-homogeneous space that is not σ-homogeneous with pairwise disjoint witnesses? At least under additional set-theoretic assumptions? References

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Kurt Godel¨ Research Center for Mathematical Logic Institute of Mathematics, University of Vienna Kolingasse 14-16 1090 Vienna, Austria Email address: [email protected] URL: http://www.logic.univie.ac.at/~medinia2/

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