A new topological generalization of descriptive set theory

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.