# Generalized Choquet spaces

With Philipp Schlicht. *Fundamenta mathematica*, 2016. (arχiv | journal)</p>

*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.

*In more detail*: Descriptive set theorists work with Polish topological spaces, that is, spaces that are second countable and completely metrizable. Every uncountable Polish space is isomorphic via a Borel bijection to the very special Baire space $\omega^\omega$.

A lot of work has been dedicated to adapting results from descriptive set theory to higher cardinals, where it is natural to work with the generalization of the Baire space $\kappa^\kappa$. Here we use the $<\kappa$-supported topology, so if $\kappa^{<\kappa}=\kappa$ then the Baire space has weight $\kappa$.

In order to generalize the notion of Polish space or standard Borel space to the setting of spaces of weight $\kappa$, it is necessary to find a replacement for the notion of complete metrizability. In this article we consider spaces in which player II has a winning strategy in a $\kappa$-length variant of the strong Choquet game. We show for instance that under appropriate hypotheses, every such space of size $>\kappa$ is isomorphic via a $\kappa$-Borel bijection to the generalized Baire space $\kappa^\kappa$.

Later in the article, we study the generalization of Polish metric spaces where ordinary metrics are replaced by metrics that take values in a set of size $\kappa$. (Such spaces have been studied extensively, as well as much further generalizations.) We show that there exists a universal Uryoshn space of this type. We also show that Debs’ characterization of hereditarily Baire spaces can be generalized to this context.