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Classical Descriptive Set Theory

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Aspects of topology: On dimension theory

Geometry of sets and measures in euclidean spaces fractals and rectifiability

Fundamenta Mathematicae私抄 : 退任の辞に代えて

Descriptive Set Theory

L U C A M O T T O R O S, P H I L I P P S C H L I C H T and V I C T O R S E L I V A N O V3† 1 Abteilung fur Mathematische Logik, Mathematisches Institut, Albert-Ludwigs-Universitat Freiburg, Eckerstrase 1, D-79104 Freiburg im Breisgau, Germany. luca.motto.ros@math.uni-freiburg.de 2 Mathematisches Institut, Universitat Bonn, Endenicher Allee 60, 53115 Bonn, Germany. schlicht@math.uni-bonn.de 3 A. P. Ershov Institute of Informatics Systems, Siberian Division Russian Academy of Sciences, Novosibirsk, Russia. vseliv@iis.nsk.su

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Wadge-like reducibilities on arbitrary quasi-Polish spaces

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.

By classical results of Hurewicz, Kechris and Saint-Raymond, an analytic subset of a Polish space X is covered by a Kσ subset of X if and only if it does not contain a closed-in-X subset homeomorphic to the Baire space ww. We consider the analogous statement (which we call the Hurewicz dichotomy) for ∑11j subsets of the generalized Baire space κκ for a given uncountable cardinal κ with κ = κ<κ. We show that the statement that this dichotomy holds at all uncountable regular cardinals is consistent with the axioms of ZFC together with GCH and large cardinal axioms. In contrast, we show that the dichotomy fails at all uncountable regular cardinals after we add a Cohen real to a model of GCH. We also discuss connections with some regularity properties, like the κ-perfect set property, the κ-Miller measurability, and the κ-Sacks measurability.

/pdf/the-hurewicz-dichotomy-for-generalized-baire-spaces-3bhqp6mzri.pdf

The Hurewicz dichotomy for generalized Baire spaces

https://www.math.uni-bonn.de/people/schlicht/Thin_equivalence_relations_and_inner_models.pdf

Thin equivalence relations and inner models

Let κ be an uncountable cardinal with \(\kappa = {\kappa ^{ {\kappa ^ + }\)” implies the statement “every closed subset of \(^\kappa \kappa \) is a continuous image of \({}^\kappa ({\kappa ^ + })\)” or its negation.

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Philipp Schlicht

Papers

Wadge-like reducibilities on arbitrary quasi-Polish spaces

Wadge-like reducibilities on arbitrary quasi-Polish spaces

The Hurewicz dichotomy for generalized Baire spaces

Thin equivalence relations and inner models

Continuous images of closed sets in generalized Baire spaces