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Borel-amenable reducibilities for sets of reals

Luca Motto Ros

Journal of Symbolic Logic 74 (1):27-49 (2009)

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Every zero-dimensional homogeneous space is strongly homogeneous under determinacy.Raphaël Carroy, Andrea Medini & Sandra Müller - 2020 - Journal of Mathematical Logic 20 (3):2050015.details 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 (i.e. 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), complements a result of van Douwen, (...) Download Export citation Bookmark 2 citations
Continuous reducibility and dimension of metric spaces.Philipp Schlicht - 2018 - Archive for Mathematical Logic 57 (3-4):329-359.details If is a Polish metric space of dimension 0, then by Wadge’s lemma, no more than two Borel subsets of X are incomparable with respect to continuous reducibility. In contrast, our main result shows that for any metric space of positive dimension, there are uncountably many Borel subsets of that are pairwise incomparable with respect to continuous reducibility. In general, the reducibility that is given by the collection of continuous functions on a topological space \\) is called the Wadge quasi-order (...) Download Export citation Bookmark 2 citations
Baire reductions and good Borel reducibilities.Luca Motto Ros - 2010 - Journal of Symbolic Logic 75 (1):323-345.details In [9] we have considered a wide class of "well-behaved" reducibilities for sets of reals. In this paper we continue with the study of Borel reducibilities by proving a dichotomy theorem for the degree-structures induced by good Borel reducibilities. This extends and improves the results of [9] allowing to deal with a larger class of notions of reduction (including, among others, the Baire class ξ functions). Download Export citation Bookmark 6 citations
Beyond Borel-amenability: scales and superamenable reducibilities.Luca Motto Ros - 2010 - Annals of Pure and Applied Logic 161 (7):829-836.details We analyze the degree-structure induced by large reducibilities under the Axiom of Determinacy. This generalizes the analysis of Borel reducibilities given in Alessandro Andretta and Donald A. Martin [1], Luca Motto Ros [6] and Luca Motto Ros. [5] e.g. to the projective levels. Download Export citation Bookmark 4 citations
Game representations of classes of piecewise definable functions.Luca Motto Ros - 2011 - Mathematical Logic Quarterly 57 (1):95-112.details We present a general way of defining various reduction games on ω which “represent” corresponding topologically defined classes of functions. In particular, we will show how to construct games for piecewise defined functions, for functions which are pointwise limit of certain sequences of functions and for Γ-measurable functions. These games turn out to be useful as a combinatorial tool for the study of general reducibilities for subsets of the Baire space [10]. Download Export citation Bookmark 6 citations
Beyond Borel-amenability: Scales and superamenable reducibilities.L. Ros - 2010 - Annals of Pure and Applied Logic 161 (7):829-836.details We analyze the degree-structure induced by large reducibilities under the Axiom of Determinacy. This generalizes the analysis of Borel reducibilities given in Alessandro Andretta and Donald A. Martin [1], Luca Motto Ros [6] and Luca Motto Ros. [5] e.g. to the projective levels. Download Export citation Bookmark 3 citations
Constructing wadge classes.Raphaël Carroy, Andrea Medini & Sandra Müller - 2022 - Bulletin of Symbolic Logic 28 (2):207-257.details We show that, assuming the Axiom of Determinacy, every non-selfdual Wadge class can be constructed by starting with those of level $\omega _1$ and iteratively applying the operations of expansion and separated differences. The proof is essentially due to Louveau, and it yields at the same time a new proof of a theorem of Van Wesep. The exposition is self-contained, except for facts from classical descriptive set theory. Download Export citation Bookmark 1 citation

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