Generalized Choquet spaces

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.