Infinite Powers and Cohen Reals

Medini, Andrea; Mill, Jan van; Zdomskyy, Lyubomyr

doi:10.4153/cmb-2017-055-x

Cited by 3 publications

(1 citation statement)

References 7 publications

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“…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 ω 1 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.…”

Section: A Counterexample In Zfcmentioning

confidence: 99%

Zero-dimensional $σ$-homogeneous spaces

Medini¹,

Vidnyánszky²

2021

Preprint

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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 alternative proofs of results of van Engelen, Miller and Steel. It is an open problem whether every analytic zero-dimensional space is σ-homogeneous. Contents 17 13. Final remarks and open questions 19 References 21

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Section: A Counterexample In Zfcmentioning

confidence: 99%

Zero-dimensional $σ$-homogeneous spaces

Medini¹,

Vidnyánszky²

2021

Preprint

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View full text Add to dashboard Cite

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Zero-dimensional σ-homogeneous spaces

Medini

Vidnyánszky

2024

Annals of Pure and Applied Logic

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Every zero-dimensional homogeneous space is strongly homogeneous under determinacy

2020

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All spaces are assumed to be separable and metrizable. We show that, assuming the Axiom of Determinacy, every zero-dimensional homogeneous space is strongly homogeneous (that is, all its non-empty clopen subspaces are homeomorphic), with the trivial exception of locally compact spaces. In fact, we obtain a more general result on the uniqueness of zero-dimensional homogeneous spaces which generate a given Wadge class. This extends work of van Engelen (who obtained the corresponding results for Borel spaces) and complements a result of van Douwen. . The authors are grateful to Alessandro Andretta for valuable bibliographical help, and to Alain Louveau for allowing them to use his unpublished book [Lo2].2 RAPHAËL CARROY, ANDREA MEDINI, AND SANDRA MÜLLER Theorem 1.1. Assume AD. If X is a zero-dimensional homogeneous space that is not locally compact then X is strongly homogeneous.The above theorem follows from a uniqueness result about zero-dimensional homogeneous spaces, namely Theorem 15.2, which is of independent interest. This theorem essentially states that, for every sufficiently high level of complexity Γ, there are at most two homogeneous zero-dimensional spaces of complexity exactly Γ (one meager and one Baire).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. In fact, most of this article (Sections 3 to 13) is purely Wadge-theoretic in character. The ultimate goal of the Wadge-theoretic portion of the paper is to show that good Wadge classes are closed under intersection with Π 0 2 sets (see Section 12), hence they are reasonably closed (see Section 13). Homogeneity comes into play in Section 14, where we show that rXs is a good Wadge class whenever X is a homogeneous space of sufficiently high complexity. This will allow us to use a theorem of Steel from [St2], which will in turn yield the uniqueness result mentioned above (see Section 15). In the preceding sections, the necessary tools are developed. More specifically, Section 4 is devoted to the analysis of the selfdual Wadge classes, Sections 5 to 8 develop the machinery of relativization through Hausdorff operations, and Sections 9 to 11 develop the notions of level and expansion.The application of Wadge theory to the study of homogeneous spaces was pioneered by van Engelen in [vE3], who obtained the classification mentioned above. As a corollary (namely,[vE3, Corollary 4.4.6]), he obtained the Borel version of Theorem 1.1. The reason why his results are limited to Borel spaces is that they are all based on the fine analysis of the Borel Wadge classes given by Louveau in [Lo1]. Fully extending this analysis beyond the Borel realm appears to be a very hard problem (although partial results have been obtained in [Fo]). Here, we will follow a different strategy, and we will "substitute" facts from [Lo1] about Borel Wadge classes with more general results about arbitrary Wadge classes (under AD). Furthermore, since most of ...

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Infinite Powers and Cohen Reals

Cited by 3 publications

References 7 publications

Zero-dimensional $σ$-homogeneous spaces

Zero-dimensional $σ$-homogeneous spaces

Zero-dimensional σ-homogeneous spaces

Every zero-dimensional homogeneous space is strongly homogeneous under determinacy

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