Generalized Choquet spaces

Samuel Coskey, and Philipp Schlicht. "Generalized Choquet spaces." Fundamenta Mathematicae 232.3 (2016): 227-248. <http://eudml.org/doc/286328>.

@article{SamuelCoskey2016,
abstract = {We introduce an analog to the notion of Polish space for spaces of weight ≤ κ, where κ is an uncountable regular cardinal such that $κ^\{<κ\} = κ$. Specifically, we consider spaces in which player II has a winning strategy in a variant of the strong Choquet game which runs for κ 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 $2^\{κ\}$ many such spaces up to homeomorphism. We also establish a Kuratowski-like theorem that under mild hypotheses, any two such spaces of size > κ are isomorphic by a κ-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 κ. 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 κ-Baire.},
author = {Samuel Coskey, Philipp Schlicht},
journal = {Fundamenta Mathematicae},
keywords = {descriptive set theory; Choquet game; Choquet space; Urysohn space},
language = {eng},
number = {3},
pages = {227-248},
title = {Generalized Choquet spaces},
url = {http://eudml.org/doc/286328},
volume = {232},
year = {2016},
}

TY - JOUR
AU - Samuel Coskey
AU - Philipp Schlicht
TI - Generalized Choquet spaces
JO - Fundamenta Mathematicae
PY - 2016
VL - 232
IS - 3
SP - 227
EP - 248
AB - We introduce an analog to the notion of Polish space for spaces of weight ≤ κ, where κ is an uncountable regular cardinal such that $κ^{<κ} = κ$. Specifically, we consider spaces in which player II has a winning strategy in a variant of the strong Choquet game which runs for κ 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 $2^{κ}$ many such spaces up to homeomorphism. We also establish a Kuratowski-like theorem that under mild hypotheses, any two such spaces of size > κ are isomorphic by a κ-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 κ. 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 κ-Baire.
LA - eng
KW - descriptive set theory; Choquet game; Choquet space; Urysohn space
UR - http://eudml.org/doc/286328
ER -