Quasi-Polish Spaces

TL;DR

This paper explores the properties and characterizations of quasi-Polish spaces, a generalization of Polish spaces to non-Hausdorff settings, revealing their structure through descriptive set theory, domain theory, and effectivity.

Contribution

It introduces quasi-Polish spaces, characterizes them via Borel hierarchy and domain theory, and extends classical results of Polish spaces to this broader class.

Findings

Subspaces of quasi-Polish spaces are quasi-Polish iff they are level _2 in the Borel hierarchy.

Every -continuous domain is quasi-Polish.

The Borel hierarchy does not collapse on uncountable quasi-Polish spaces.

Abstract

We investigate some basic descriptive set theory for countably based completely quasi-metrizable topological spaces, which we refer to as quasi-Polish spaces. These spaces naturally generalize much of the classical descriptive set theory of Polish spaces to the non-Hausdorff setting. We show that a subspace of a quasi-Polish space is quasi-Polish if and only if it is level \Pi_2 in the Borel hierarchy. Quasi-Polish spaces can be characterized within the framework of Type-2 Theory of Effectivity as precisely the countably based spaces that have an admissible representation with a Polish domain. They can also be characterized domain theoretically as precisely the spaces that are homeomorphic to the subspace of all non-compact elements of an \omega-continuous domain. Every countably based locally compact sober space is quasi-Polish, hence every \omega-continuous domain is quasi-Polish. A metrizable space is quasi-Polish if and only if it is Polish. We show that the Borel hierarchy on an uncountable quasi-Polish space does not collapse, and that the Hausdorff-Kuratowski theorem generalizes to all quasi-Polish spaces.