36th Summer Topology Conference

July 18-22, 2022

University of Vienna, Department of Mathematics
Oskar-Morgenstern-Platz 1, 1090 Vienna, AUSTRIA

"New Topological Generalizations of Descriptive Set Theory and Applications to Selection Principles"

Tall, Franklin

This is joint work with Ivan Ongay Valverde. Descriptive Set Theory studies "definable" sets of reals and has many applications. Classically, Borel and analytic sets were studied, but the theory became difficult when people tried to extend it to the projective sets. However, determinacy axioms following from large cardinals gave a nice theory for the projective and even $\sigma$-projective sets. Recently, some set theorists have been generalizing Descriptive Set Theory by replacing $^{\omega}\omega$ (i.e., the space of irrationals) by $^{\kappa}\kappa$ for a regular cardinal $\kappa$. There are other generalizations, dating from the 1960's, which are topological in nature. In particular, several researchers defined $K$-$\mathit{analytic}$ spaces in various equivalent ways, e.g. upper semi-continuous compact-valued images of the space of irrationals or continuous images of Lindelöf Cech-complete spaces. These spaces have found many applications in Functional Analysis. We study several different generalizations of $K$-analytic spaces, e.g. replacing the irrationals by $\sigma$-projective sets and assuming the Axiom of $\sigma$-Projective Determinacy. The $\mathit{Definable}\, \mathit{Menger}\, \mathit{Conjecture}$ asserts that definable Menger spaces are $\sigma$-compact. A weaker version---true for example for Menger $K$-analytic spaces---is that they are Hurewicz. We investigate these conjectures for various of these generalizations, proving positive results, sometimes with the aid of determinacy axioms.

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