On the definability of Menger spaces which are not /sigma-compact

TL;DR

This paper investigates the conditions under which Menger spaces are sigma-compact, extending known results to broader classes and exploring consistency results related to definability and compactness.

Contribution

It extends Hurewicz's theorem to Cech-complete Menger spaces and explores the consistency of non-sigma-compact Menger spaces within projective and co-analytic classes.

Findings

Cech-complete Menger spaces are sigma-compact

Consistent existence of co-analytic Menger spaces that are not sigma-compact

Extension of results to projective Menger spaces under certain set-theoretic assumptions

Abstract

Hurewicz proved completely metrizable Menger spaces are /sigma-compact. We extend this to Cech-complete Menger spaces and consistently to projective Menger metrizable spaces. On the other hand, it is consistent that there is a co-analytic Menger space that is not /sigma-compact.