We introduce an analogue to Polish spaces for uncountable regular cardinals \(\kappa\) with \(\kappa = \kappa\) via a variant of the Choquet game of length \(\kappa\). There is a surjectively universal such space, and any two such spaces of size \(> \kappa\) with no points which are the intersection of fewer than \(\kappa\) open sets are \(\kappa\)-Borel isomorphic. We consider the special case of generalized \(\kappa\)-valued ultrametric spaces with the property that the intersection of any decreasing sequence of balls is nonempty and construct a family of universal Urysohn spaces. We then prove that the logic action of Sym(\(\kappa\)) is universal for \(\kappa\)-Borel measurable actions of Sym(\(\kappa\)) with respect to equivariant embeddings.
This is joint work with Samuel Coskey.