However, Polish spaces are mostly studied today because they are the primary setting for descriptive set theory, including the study of Borel equivalence relations.
A separable metric space is completely metrizable if and only if the second player has a winning strategy in this game.
It states that a separable metric space is completely metrizable if and only if it is a
There are several classic results of Banach, Freudenthal and Kuratowski on homomorphisms between Polish groups.
[10] Firstly, Banach's argument[11] applies mutatis mutandis to non-Abelian Polish groups: if G and H are separable metric spaces with G Polish, then any Borel homomorphism from G to H is continuous.
[12] Secondly, there is a version of the open mapping theorem or the closed graph theorem due to Kuratowski:[13] a continuous injective homomorphism of a Polish subgroup G onto another Polish group H is an open mapping.
As a result, it is a remarkable fact about Polish groups that Baire-measurable mappings (i.e., for which the preimage of any open set has the property of Baire) that are homomorphisms between them are automatically continuous.