Borel–Wadge degrees

Andretta, Alessandro; Martin, Donald A.

doi:10.4064/fm177-2-5

Cited by 24 publications

(40 citation statements)

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“…Andretta (2006), Andretta and Martin (2003), Motto Ros (2009), Motto Ros (2010a) and Motto Ros (2010b)). In this section, we will provide several examples which are relevant for this paper.…”

Section: Some Examples Of Reducibilitiesmentioning

confidence: 94%

Wadge-like reducibilities on arbitrary quasi-Polish spaces

Ros¹,

Schlicht²,

Selivanov

2014

Math. Struct. Comp. Sci.

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The structure of the Wadge degrees on zero-dimensional spaces is very simple (almost well-ordered), but for many other natural non-zero-dimensional spaces (including the space of reals) this structure is much more complicated. We consider weaker notions of reducibility, including the so-called \Delta^0_\alpha-reductions, and try to find for various natural topological spaces X the least ordinal \alpha_X such that for every \alpha_X \leq \beta < \omega_1 the degree-structure induced on X by the \Delta^0_\beta-reductions is simple (i.e. similar to the Wadge hierarchy on the Baire space). We show that \alpha_X \leq {\omega} for every quasi-Polish space X, that \alpha_X \leq 3 for quasi-Polish spaces of dimension different from \infty, and that this last bound is in fact optimal for many (quasi-)Polish spaces, including the real line and its powers.Comment: 50 pages, revised version, accepted for publication on Mathematical Structures in Computer Scienc

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Section: Some Examples Of Reducibilitiesmentioning

confidence: 94%

Wadge-like reducibilities on arbitrary quasi-Polish spaces

Ros¹,

Schlicht²,

Selivanov

2014

Math. Struct. Comp. Sci.

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“…This is a well-known result (see for example [Lo2,Lemma 7.3.4]). Our approach is essentially the same as the one used in the proof of [AM,Theorem 16] or in [MR,Theorem 5.3]. However, since the proof becomes slightly simpler in our context, we give all the details below.…”

Section: The Analysis Of Selfdual Setsmentioning

confidence: 99%

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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“…1 As usual in Descriptive Set Theory we will always identify R with the Baire space ω ω. 2. if λ is an ordinal of uncountable cofinality and ξ | ξ < λ is a chain of tractable pointclasses (i.e.…”

Section: Introductionmentioning

confidence: 99%

“…Building on the work of Andretta and Martin in [1] (where the case when F is the set of all Borel functions was considered), in [6] and [5] we have investigated various reducibility notions in the Borel context, but it is clear that there are also some natural sets of functions (such as projective functions) that can be used as reductions and which are strictly larger than the set of Borel functions. In this paper we will prove that, assuming AD+DC, structural results similar to those for the Borel context can be proved for larger and larger pointclasses.…”

Section: Introductionmentioning

confidence: 99%