Definable versions of Menger's conjecture

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.