Detecting  σZn-sets in topological groups and linear metric spaces

Taras Banakh


We prove that if an analytic subset A of a linear metric space X is not contained in a σZω-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 X of X. This implies two results:
(i) an analytic subgroup A of a linear metric space X is a σZω-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 σZω-space  if A contains no open Polish subspace and A contains a Polish convex set K with dense affine hull in X.

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