Zero-Dimensional $\Sigma $-Homogeneous Spaces

$Zero-Dimensional $\Sigma $-Homogeneous Spaces$ ZERO-DIMENSIONAL σ-HOMOGENEOUS SPACES ANDREA MEDINI AND ZOLTAN´ VIDNYANSZKY´ Dedicated to the memory of Ken Kunen Abstract. All spaces are assumed to be separable and metrizable. Ostrovsky showed that every zero-dimensional Borel space is σ-homogeneous. Inspired by this theorem, we obtain the following results: ‚ Assuming AD, every zero-dimensional space is σ-homogeneous, ‚ Assuming AC, there exists a zero-dimensional space that is not σ-homogeneous, ‚ Assuming V “ L, there exists a coanalytic zero-dimensional space that is not σ-homogeneous. Along the way, we introduce two notions of hereditary rigidity, and give alter- native proofs of results of van Engelen, Miller and Steel. It is an open problem whether every analytic zero-dimensional space is σ-homogeneous. Contents 1. Introduction 1 2. Preliminaries and notation 3 3. The basics of Wadge theory 6 4. Relativization 7 5. The notion of level 8 6. Steel’stheoremandgoodWadgeclasses 9 7. Zero-dimensionalhomogeneousspacesoflowcomplexity 9 8. The positive result 12 9. Hereditarily rigid spaces 14 10. A counterexample in ZFC 15 11. Preliminaries on V “ L 16 12. Definable counterexamples under V “ L 17 13. Final remarks and open questions 19 arXiv:2107.07747v1 [math.GN] 16 Jul 2021 References 21 1. Introduction Throughout this article, unless we specify otherwise, we will be working in the theory ZF ` DC, that is, the usual axioms of Zermelo-Fraenkel (without the Axiom Date: July 14, 2021. 2010 Mathematics Subject Classification. 54H05, 03E15, 03E60. Key words and phrases. Homogeneous, zero-dimensional, determinacy, V “ L, Wadge theory, constructible, rigid. The first-listed author was supported by the FWF grant P 30823. The second-listed author was partially supported by the National Research, Development and Innovation Office – NKFIH grants no. 113047, no. 129211 and FWF grant M 2779. 1 2 ANDREAMEDINIANDZOLTAN´ VIDNYANSZKY´ of Choice) plus the principle of Dependent Choices (see Section 2 for more details). 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 famil- iarity with the basic theory of Borel, analytic and coanalytic sets in Polish spaces, and mostly use the same notation as in [Ke].

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