Generalized Choquet spaces

Samuel Coskey ¹   ; Philipp Schlicht ²

¹ Department of Mathematics Boise State University 1910 University Drive Boise, ID 83725-1555, U.S.A.
² Mathematisches Institut Universität Bonn Endenicher Allee 60 53115 Bonn, Germany

Fundamenta Mathematicae, Tome 232 (2016) no. 3, pp. 227-248

Résumé

We introduce an analog to the notion of Polish space for spaces of weight $\leq \kappa $, where $\kappa $ is an uncountable regular cardinal such that $\kappa ^{\kappa }=\kappa $. Specifically, we consider spaces in which player II has a winning strategy in a variant of the strong Choquet game which runs for $\kappa $ 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^\kappa $ many such spaces up to homeomorphism. We also establish a Kuratowski-like theorem that under mild hypotheses, any two such spaces of size $>\kappa $ are isomorphic by a $\kappa $-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 $\kappa $. 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 $\kappa $-Baire.

Détail

DOI : 10.4064/fm924-12-2015

Keywords: introduce analog notion polish space spaces weight leq kappa where kappa uncountable regular cardinal kappa kappa kappa specifically consider spaces which player nbsp has winning strategy variant strong choquet game which runs kappa many rounds after discussing basic theory these games spaces prove there surjectively universal space there exactly kappa many spaces homeomorphism establish kuratowski like theorem under mild hypotheses spaces size kappa isomorphic kappa borel function consider dynamic version choquet game existence winning strategy player nbsp implies existence winning tactic strategy depends only recent move study generalization polish ultrametric spaces where ultrametric allowed values set size kappa context there family universal urysohn type spaces characterization spaces which hereditarily kappa baire