Generalized Choquet spaces

作者：Coskey Samuel^*; Schlicht Philipp

来源：Fundamenta Mathematicae, 2016, 232(3): 227-248.

DOI：10.4064/fm924-12-2015

• descriptive set theory
• Choquet game
• Choquet space
• Urysohn space

We introduce an analog to the notion of Polish space for spaces of weight <= kappa, where n 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 n 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 > n are isomorphic by a n-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 n. 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.

• 出版日期2016