Generalized Choquet spaces

• Autorzy: Samuel Coskey , Philipp Schlicht
• Języki publikacji: EN

Abstrakty

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.

Kategorie tematyczne

• 91A44: Games involving topology or set theory
• 03E15: Descriptive set theory

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Fundamenta Mathematicae
• Rocznik: 2016
• Tom: 232
• Numer: 3
• Strony: 227-248

Daty

wydano

2016

Twórcy

autor

Samuel Coskey

• Department of Mathematics, Boise State University, 1910 University Drive, Boise, ID 83725-1555, U.S.A.

autor

Philipp Schlicht

• Mathematisches Institut, Universität Bonn, Endenicher Allee 60, 53115 Bonn, Germany

Identyfikatory

DOI

10.4064/fm924-12-2015