Master Thesis Descriptive Set Theory Continuous Reductions on Polish Spaces and Quasi-Polish Spaces Louis Vuilleumier Under the supervision of Pr. Jacques Duparc January 15, 2016 Abstract A subset A ⊆ X of a topological space X is continuously reducible to a subset 1 B ⊆ X if there exists a continuous function f ∶ X → X such that f − (B) = A: In this case, we write A ≤W B: The binary relation ≤W is a quasi-order. This Master Thesis is devoted to the study of the continuous reducibility for subsets of some topological spaces. First, we study continuous reducibility on the Baire ! space ! using games. Then, we show that ≤W is very complicated on the subsets of the real line. Finally, we give a game characterization of ≤W in P(!) equipped with the Scott topology. Contents Acknowledgments 1 Introduction 2 1 Prerequisites 4 1.1 Set Theory . .4 1.2 Topology . .5 1.3 Metric Spaces . .7 1.4 Trees . .9 1.5 Infinite Sequences . 10 1.6 Baire Property . 11 2 Polish Spaces 15 2.1 Definition and Examples . 15 2.2 Polish Subspaces . 19 2.3 Some Universal Properties . 23 2.4 Perfect Polish Spaces . 23 2.5 Zero-Dimensional Polish Spaces . 31 3 Hierarchies on Polish Spaces 34 3.1 Borel Hierarchy . 34 3.2 Difference Hierarchy . 43 3.3 Lipschitz and Wadge Hierarchies . 44 4 Games and Hierarchies 48 4.1 Gale-Stewart Games on !! ....................... 48 4.2 Lipschitz and Wadge Games on !! .................. 53 4.3 Lipschitz and Wadge Hierarchy on !! ................ 58 4.4 Construction of the Lower Levels of the Wadge Hierarchy of !! . 70 4.5 Cantor Space . 76 5 Wadge Hierarchy on R 78 6 Quasi-Polish Spaces 86 6.1 Definition . 86 6.2 Examples . 88 6.3 Borel Hierarchy and Universal Property of P(!) .......... 91 6.4 Two Views on the Wadge Hierarchy on P(!) ............ 95 6.5 Games on P(!) .............................. 98 6.6 First Results . 104 Conclusion 107 Acknowledgments First, I would like to thank Pr. Jacques Duparc who allowed me to do this Master Thesis with him. I also thank him for the exciting hours he took to allow me to enter in this new world of descriptive set theory. Then, I am especially grateful to Gianluca Basso who agreed to take some of his time to correct lots of mistakes in my work with an exceptional enthusiasm. Without him, this Master Thesis would not be as readable as it is. I would also like to thank Kevin Fournier and Yann Pequignot who took some time to clarify some mathematical points which were confusing to me and to share some of their experience with me. As this Master Thesis is also the final point of my student life, I have a grateful thought to all my friends with whom I spent wonderful years of study. I start by William Borgeaud which, in addition to being my friend, helped me with some proofs of this work. Then, I think of Aurélien Chappuis, Léo Diserens, Gaëtan Gogniat, Clélia Liebermann, Nicolas Rossi, Matthieu Wimmer and all the others that I cannot list here. Thank you to all of you ! Finalement, je tiens à remercier toute ma famille, et tout particulièrement mes parents. Maman, Papa, plus que de m’en donner la possibilité, vous m’avez donné l’envie de comprendre le monde qui nous entoure, ainsi que tout le soutien que j’aurais pu espérer. Merci. January 15, 2016, Louis Vuilleumier 1 Introduction Historically, descriptive set theory is the study of the subsets of the real line. It was born with measure theory, when the question of which subsets of R are measurable, and which are not, was addressed. During the development of this new area of study, mathematicians observed that they could generalize this study to the study of subsets of more general spaces, the Polish spaces, without losing too many tools. In order to understand the measurability of subsets, it is an usual practice to hierarchize them from the simple ones to the more complex ones. A first way of doing this is to consider the σ-algebra of the Borel sets, with the hierarchy obtained by considering how far sets are from being open, i.e. which operations we need to construct them from the open sets. This hierarchy is fundamental in the development of descriptive set theory. One interesting property of each class of this hierarchy is the closure under continuous preimages. Therefore, W. Wadge [Wad84] had the idea of consid- ering the hierarchy built from classes closed under continuous preimages. It is the Wadge hierarchy. This is natural in a topological sense since all classical topology is built from the open sets and the continuous functions. We can view two subsets A and B in some topological space X as being equivalent in the 1 Wadge hierarchy if there exist two functions f; g ∶ X → X such that f − (A) = B 1 and g− (B) = A. In this case, we say that A is continuously reducible to B and vice versa. Wadge’s famous idea was to consider games that are equivalent to continuous reducibility in the Baire space !! of infinite sequences of integers. Using game theoretical tools, Wadge was able to give a whole, nice picture of the Wadge hierarchy on the Borel sets of the Baire space !!. The first part of this Master Thesis is to try to understand the Wadge hierarchy on some Polish spaces and to try to justify the constructions and the choices which are made all along the study. A second part consists of trying to study a similar game characterization of continuous reducibility designed by Pr. ! Jacques Duparc on a space which is different from ! ; namely P(!) equipped with the Scott topology. In Chapter 1, we give some of the set theoretical and topological prerequisites and notations that we will need all along the work. It is in this Chapter that we define the most important space of Wadge’s work, the Baire space !!. In Chapter 2, we define and understand the objects of study of descriptive set theory, the Polish spaces. We start by giving examples and then we give some properties of these spaces. One important property is the universality of !! among all the zero-dimensional Polish spaces (Theorem 2.54), which are a kind of Polish spaces. Indeed, this result allows us to design games in all zero-dimensional Polish spaces. Chapter 3 is dedicated to the definitions of hierarchies on Polish spaces, 2 such as the Borel hierarchy and the Wadge hierarchy. In this Chapter, we try to motivate our deep study of the Wadge hierarchy on !!. In Chapter 4, we come finally to the definition of the game designed by Wadge. With this game and some game theoretical tools such as Borel-determinacy (Theorem 4.10), we obtain the whole picture of the Wadge hierarchy on the Borel subsets of the Baire space. In particular, we show that this hierarchy is well-founded and has maximal antichains of size 2. Chapter 5 is dedicated to the study of the Wadge hierarchy on the real line. In this Chapter, we show that, without a game characterization, the Wadge hierarchy becomes very difficult to understand. In particular, it is not well- founded and it has infinite antichains. Finally, in Chapter 6, we try to generalize Polish spaces by considering quasi- Polish spaces. M. De Brecht [DB13] showed that it is an interesting generaliza- tion for several reasons. Among these quasi-Polish spaces, P(!) equipped with the Scott topology is universal (Theorem 6.30). The study of the Wadge hier- archy on this space is encouraged by the game characterization of continuous function designed by Pr. Jacques Duparc. Hence, we hope this game will give us a nice picture of the Wadge hierarchy of P(!). This study that we constructed in this work is not exhaustive and deserves more time. 3 Chapter 1 Prerequisites In this Chapter, we introduce all the basic notions we need in the rest of this work. This is mostly notations and well known topological facts. The major part of this Chapter is covered in a first semester course in Topology, hence, we will most of the time omit the proofs. 1.1 Set Theory We precise that, all along this work, our framework is classical logic and we work within the theory (ZF ) of Zermelo-Fraenkel. Sometimes, we will need more assumptions on the theory, such as the Axiom of Choice (AC), or the Axiom of countable Choice (AC!). We just recall the definitions of the Axiom of Choice and of countable Choice. The Axiom of Choice guarantees the existence of a choice function on every family of nonempty sets. Hence, we can state it as: ∀S(X ∈ S → ∅ ≠ X) → ∃f ∶ S → X; (f(X) ∈ X) : X∈S The Axiom of countable Choice is the same statement applied only for countable families S: Under (AC!); we can prove the following result which we will use a lot. Proposition 1.1. A countable union of countably many sets is countable. In the rest of this work, we will always assume the Axiom of countable Choice to be true. In some cases, for example to construct pathological behaviors, we will use the Axiom of Choice. Hence, in these cases, we will say explicitly that we assume the Axiom of Choice. We have two others characterizations of the Axiom of Choice. First, it is equivalent to Zermelo’s Theorem. Theorem 1.2. Every set can be well-ordered. A well-ordering of a set is a binary relation which is total, anti-symmetric, transitive and such that every nonempty subset has a least element.