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GAMES AND HEREDITARY BAIRENESS IN HYPERSPACES AND SPACES OF PROBABILITY MEASURES

Part of: Basic constructions Game theory Probability theory on algebraic and topological structures Fairly general properties Spaces with richer structures

Published online by Cambridge University Press:  15 June 2020

Mikołaj Krupski Open the ORCID record for Mikołaj Krupski [Opens in a new window]
Mikołaj Krupski*
Affiliation:
Institute of Mathematics, University of Warsaw, ul. Banacha 2, 02–097 Warszawa, Poland (mkrupski@mimuw.edu.pl)
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Abstract

We establish that the existence of a winning strategy in certain topological games, closely related to a strong game of Choquet, played in a topological space $X$ and its hyperspace $K(X)$ of all nonempty compact subsets of $X$ equipped with the Vietoris topology, is equivalent for one of the players. For a separable metrizable space $X$, we identify a game-theoretic condition equivalent to $K(X)$ being hereditarily Baire. It implies quite easily a recent result of Gartside, Medini and Zdomskyy that characterizes hereditary Baire property of hyperspaces $K(X)$ over separable metrizable spaces $X$ via the Menger property of the remainder of a compactification of $X$. Subsequently, we use topological games to study hereditary Baire property in spaces of probability measures and in hyperspaces over filters on natural numbers. To this end, we introduce a notion of strong $P$-filter ${\mathcal{F}}$ and prove that it is equivalent to $K({\mathcal{F}})$ being hereditarily Baire. We also show that if $X$ is separable metrizable and $K(X)$ is hereditarily Baire, then the space $P_{r}(X)$ of Borel probability Radon measures on $X$ is hereditarily Baire too. It follows that there exists (in ZFC) a separable metrizable space $X$, which is not completely metrizable with $P_{r}(X)$ hereditarily Baire. As far as we know, this is the first example of this kind.

Keywords

hyperspaceVietoris topologyhereditarily Baire spacestrong Choquet gamePorada gameMenger spaceMenger at infinity spacefilterstrong P-filterspace of probability measuresProhorov space

MSC classification

Primary: 54B20: Hyperspaces 54E52: Baire category, Baire spaces 60B05: Probability measures on topological spaces 91A44: Games involving topology or set theory
Secondary: 54D40: Remainders 54D20: Noncompact covering properties (paracompact, Lindelöf, etc.) 54D80: Special constructions of spaces (spaces of ultrafilters, etc.)

Information

Type
Research Article
Information
Journal of the Institute of Mathematics of Jussieu , Volume 21 , Issue 3 , May 2022 , pp. 851 - 868
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Copyright
© The Author(s) 2020. Published by Cambridge University Press

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