Generalized Polish spaces at regular uncountable cardinals

TL;DR

This paper develops a comprehensive framework for generalized Polish spaces at uncountable regular cardinals, extending previous work and providing new characterizations of generalized Cantor and Baire spaces.

Contribution

It introduces a unified approach to generalized Polish-like spaces and characterizes key examples, advancing the theory in the context of uncountable cardinals.

Findings

Extended previous results on generalized Polish spaces.

Provided natural characterizations of generalized Cantor and Baire spaces.

Answered open questions from prior research.

Abstract

In the context of generalized descriptive set theory, we systematically compare and analyze various notions of Polish-like spaces and standard $\kappa$-Borel spaces for $\kappa$ an uncountable (regular) cardinal satisfying $\kappa^{<\kappa} = \kappa$. As a result, we obtain a solid framework where one can develop the theory in full generality. We also provide natural characterizations of the generalized Cantor and Baire spaces. Some of the results obtained considerably extend previous work from [Coskey-Schlicht 2016, Galeotti 2019, Luecke-Schlicht 2015], and answer some questions contained therein.