Banalytic spaces and characterization of Polish groups

TL;DR

This paper introduces banalytic spaces, characterizes Polish groups using these spaces, and explores their properties and equivalences under certain set-theoretic assumptions.

Contribution

It defines banalytic spaces and establishes their role in characterizing Polish groups and related topological groups.

Findings

Banalytic spaces are images of Polish spaces under Borel maps.

Regular banalytic spaces are cosmic and have countable spread.

For Baire topological groups, several conditions are equivalent to being Polish.

Abstract

A topological space is defined to be banalytic (resp. analytic) if it is the image of a Polish space under a Borel (resp. continuous) map. A regular topological space is analytic if and only if it is banalytic and cosmic. Each (regular) banalytic space has countable spread (and under PFA is hereditarily Lindel\"of). Applying banalytic spaces to topological groups, we prove that for a Baire topological group $X$ the following conditions are equivalent: (1) $X$ is Polish, (2) $X$ is analytic, (3) $X$ is banalytic and cosmic, (4) $X$ is banalytic and has countable pseudocharacter. Under PFA the conditions (1)--(4) are equivalent to the banalycity of $X$. The conditions (1)--(3) remain equivalent for any Baire semitopological group.