A new topological generalization of descriptive set theory

TL;DR

This paper introduces a broader topological framework extending the $\sigma$-projective hierarchy beyond Polish spaces, using set-theoretic axioms like $\sigma$-projective determinacy to generalize results in descriptive set theory.

Contribution

It develops a new topological generalization of the $\sigma$-projective hierarchy that applies to more general spaces and establishes related results under set-theoretic assumptions.

Findings

Generalized results for $K$-analytic spaces hold in ZFC.

Extended theorems to more general spaces under large cardinal assumptions.

Showed that nicely defined Menger spaces are Hurewicz or $\sigma$-compact.

Abstract

We introduce a new topological generalization of the $\sigma$-projective hierarchy, not limited to Polish spaces. Earlier attempts have replaced $^{\omega}\omega$ by $^{\kappa}\kappa$, for $\kappa$ regular uncountable, or replaced countable by $\sigma$-discrete. Instead we close the usual $\sigma$-projective sets under continuous images and perfect preimages together with countable unions. The natural set-theoretic axiom to apply is $\sigma$-projective determinacy, which follows from large cardinals. Our goal is to generalize the known results for $K$-analytic spaces (continuous images of perfect preimages of $^{\omega}\omega$) to these more general settings. We have achieved some successes in the area of Selection Principles--the general theme is that nicely defined Menger spaces are Hurewicz or even $\sigma$-compact. The $K$-analytic results are true in ZFC; the more general results have consistency strength of only an inaccessible.