A generalization of a theorem of Hurewicz for quasi-Polish spaces

TL;DR

This paper generalizes Hurewicz's theorem by identifying key non-quasi-Polish spaces and characterizing co-analytic subsets of quasi-Polish spaces as either quasi-Polish or containing specific canonical spaces.

Contribution

It introduces a generalized theorem extending Hurewicz's result to quasi-Polish spaces, identifying canonical non-quasi-Polish examples.

Findings

Identification of four canonical non-quasi-Polish spaces

Generalization of Hurewicz's theorem to quasi-Polish spaces

Characterization of co-analytic subsets in these spaces

Abstract

We identify four countable topological spaces $S_2$, $S_1$, $S_D$, and $S_0$ which serve as canonical examples of topological spaces which fail to be quasi-Polish. These four spaces respectively correspond to the $T_2$, $T_1$, $T_D$, and $T_0$-separation axioms. $S_2$ is the space of rationals, $S_1$ is the natural numbers with the cofinite topology, $S_D$ is an infinite chain without a top element, and $S_0$ is the set of finite sequences of natural numbers with the lower topology induced by the prefix ordering. Our main result is a generalization of Hurewicz's theorem showing that a co-analytic subset of a quasi-Polish space is either quasi-Polish or else contains a countable $\Pi^0_2$-subset homeomorphic to one of these four spaces.