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2016 | 232 | 3 | 227-248
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Generalized Choquet spaces

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EN
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EN
We introduce an analog to the notion of Polish space for spaces of weight ≤ κ, where κ is an uncountable regular cardinal such that $κ^{<κ} = κ$. Specifically, we consider spaces in which player II has a winning strategy in a variant of the strong Choquet game which runs for κ many rounds. After discussing the basic theory of these games and spaces, we prove that there is a surjectively universal such space and that there are exactly $2^{κ}$ many such spaces up to homeomorphism. We also establish a Kuratowski-like theorem that under mild hypotheses, any two such spaces of size > κ are isomorphic by a κ-Borel function. We then consider a dynamic version of the Choquet game, and show that in this case the existence of a winning strategy for player II implies the existence of a winning tactic, that is, a strategy that depends only on the most recent move. We also study a generalization of Polish ultrametric spaces where the ultrametric is allowed to take values in a set of size κ. We show that in this context, there is a family of universal Urysohn-type spaces, and we give a characterization of such spaces which are hereditarily κ-Baire.
Słowa kluczowe
Rocznik
Tom
232
Numer
3
Strony
227-248
Opis fizyczny
Daty
wydano
2016
Twórcy
  • Department of Mathematics, Boise State University, 1910 University Drive, Boise, ID 83725-1555, U.S.A.
  • Mathematisches Institut, Universität Bonn, Endenicher Allee 60, 53115 Bonn, Germany
Bibliografia
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Bibliografia
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Identyfikator YADDA
bwmeta1.element.bwnjournal-article-doi-10_4064-fm924-12-2015
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