Detecting σZn-sets in topological groups and linear metric spaces

Анотація

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 $\overline{X}$ of $X$. This implies two results:
$\left(i\right)$ 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 $X$;
$\left(ii\right)$ 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 $X$.