2022/10/12 by Ongay-Valverde, Iván, Tall, Franklin D.
#03E15 #03E60 #54A35 #54C10 #54C60 #54D40 #FOS: Mathematics #General Topology (math.GN) #Logic (math.LO) #Primary 54H05 #Secondary 03E15. 03E55
paper · doi:10.48550/arxiv.2210.05849
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.