The Hurewicz dichotomy for generalized Baire spaces

TL;DR

This paper explores the Hurewicz dichotomy for analytic sets in generalized Baire spaces, establishing forcing methods to make the dichotomy hold at all uncountable regular cardinals under GCH, and analyzing its failure in the constructible universe.

Contribution

It introduces forcing techniques to ensure the Hurewicz dichotomy for  sets in generalized Baire spaces at all uncountable regular cardinals, extending classical results.

Findings

The Hurewicz dichotomy can be forced to hold at all uncountable regular cardinals under GCH.

In the constructible universe, the dichotomy fails at all uncountable regular cardinals.

Adding Cohen reals destroys the dichotomy in models with GCH.

Abstract

By classical results of Hurewicz, Kechris and Saint-Raymond, an analytic subset of a Polish space $X$ is covered by a $K_\sigma$ subset of $X$ if and only if it does not contain a closed-in-$X$ subset homeomorphic to the Baire space ${}^\omega \omega$. We consider the analogous statement (which we call Hurewicz dichotomy) for $\Sigma^1_1$ subsets of the generalized Baire space ${}^\kappa \kappa$ for a given uncountable cardinal $\kappa$ with $\kappa=\kappa^{<\kappa}$, and show how to force it to be true in a cardinal and cofinality preserving extension of the ground model. Moreover, we show that if the Generalized Continuum Hypothesis (GCH) holds, then there is a cardinal preserving class-forcing extension in which the Hurewicz dichotomy for $\Sigma^1_1$ subsets of ${}^\kappa \kappa$ holds at all uncountable regular cardinals $\kappa$, while strongly unfoldable and supercompact cardinals are preserved. On the other hand, in the constructible universe L the dichotomy for $\Sigma^1_1$ sets fails at all uncountable regular cardinals, and the same happens in any generic extension obtained by adding a Cohen real to a model of GCH. We also discuss connections with some regularity properties, like the $\kappa$-perfect set property, the $\kappa$-Miller measurability, and the $\kappa$-Sacks measurability.