  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.

Logic