Definable versions of Menger's conjecture

TL;DR

This paper explores the definability and complexity of Menger spaces, showing that Menger's conjecture varies with the space's definability level and providing new characterizations and results under set-theoretic assumptions.

Contribution

It demonstrates the truth and undecidability of Menger's conjecture for various definable classes of spaces and introduces new characterizations of proper K-Lusin spaces.

Findings

Menger's conjecture is true for analytic subspaces of Polish spaces.

Undecidability of Menger's conjecture for more complex definable spaces.

Productively Lindelof Cech-complete spaces are /sigma-compact if a Michael space exists.

Abstract

Menger's conjecture that Menger spaces are /sigma-compact is false; it is true for analytic subspaces of Polish spaces and undecidable for more complex definable subspaces of Polish spaces. For non-metrizable spaces, analytic Menger spaces are /sigma-compact, but Menger continuous images of co-analytic spaces need not be. The general co-analytic case is still open, but many special cases are undecidable, in particular, Menger topological groups. We also prove that if there is a Michael space, then productively Lindelof Cech-complete spaces are /sigma-compact. We also give numerous characterizations of proper K-Lusin spaces. Our methods include the Axiom of Co-analytic Determinacy, non-metrizable descriptive set theory, and Arhangel'skii's work on generalized metric spaces.