Detecting $\sigma Z_n$-sets in topological groups and linear metric spaces

TL;DR

This paper characterizes when certain analytic subsets in linear metric spaces are $\sigma Z_ ext{omega}$-sets, linking their structure to non-meagerness of sums with dense convex sets and providing criteria for $\sigma Z_ ext{omega}$-space classification.

Contribution

It establishes new conditions under which analytic subsets are $\sigma Z_ ext{omega}$-sets in linear metric spaces, connecting topological properties with algebraic sum operations.

Findings

If an analytic set is not contained in a $\sigma Z_ ext{omega}$-set, then its sum with a dense Polish convex set is non-meager.

Analytic subgroups that are not Polish and contain a dense Polish convex set are $\sigma Z_ ext{omega}$-spaces.

Dense convex analytic subsets without open Polish subspaces and containing a dense Polish convex set are $\sigma Z_ ext{omega}$-spaces.

Abstract

We prove that if an analytic subset $A$ of a linear metric space $X$ is not contained in a $\sigma Z_\omega$-subset of $X$ then for every Polish convex set $K$ with dense affine hull in $X$ the sum $A+K$ is non-meager in $X$ and the sets $A+A+K$ and $A-A+K$ have non-empty interior in the completion $\bar X$ of $X$. This implies two results: (i) an analytic subgroup $A$ of a linear metric space $X$ is a $\sigma Z_\omega$-space if $A$ is not Polish and $A$ contains a Polish convex set $K$ with dense affine hull in $X$; (ii) a dense convex analytic subset $A$ of a linear metric space $X$ is a $\sigma Z_\omega$-space if $A$ contains no open Polish subspace and $A$ contains a Polish convex set $K$ with dense affine hull in $X$.