Generalized Choquet spaces
Volume 232 / 2016
Abstract
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.
