A new topological generalization of descriptive set theory
We introduce a new topological generalization of the $σ$-projective hierarchy, not limited to Polish spaces. Earlier attempts have replaced $^ωω$ by $^κκ$, for $κ$ regular uncountable, or replaced countable by $σ$-discrete. Instead we close the usual $σ$-projective sets under continuous images and perfect preimages together with countable unions. The natural set-theoretic axiom to apply is $σ$-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 $^ωω$) 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 $σ$-compact. The $K$-analytic results are true in ZFC; the more general results have consistency strength of only an inaccessible.