2013 seminar talk: On Generalized Choquet Spaces and Groups

Talk held by Philipp Schlicht (University of Bonn, Germany) at the KGRC seminar on 2013-05-16.


We introduce an analogue to Polish spaces for uncountable regular cardinals $\kappa$ with $\kappa^{\lt\kappa} = \kappa$ 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.

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