Notes on quasi-Polish spaces

TL;DR

This paper provides a comprehensive overview of quasi-Polish spaces, highlighting their logical characterization and extending their theoretical framework with new proofs and Baire category techniques.

Contribution

It offers a self-contained exposition of quasi-Polish spaces, emphasizing their characterization as a0b0_2 subspaces and introducing new proofs and methods.

Findings

Quasi-Polish spaces are characterized as a0b0_2 subspaces of countable powers of Sierpinski space.

New proofs utilize Baire category techniques to deepen understanding.

The paper clarifies the logical structure and properties of quasi-Polish spaces.

Abstract

Quasi-Polish spaces were introduced by de Brecht as a possibly non-Hausdorff generalization of Polish spaces sharing many of their descriptive set-theoretic properties. We give a self-contained exposition of the basic theory of quasi-Polish spaces, based on their "logical" characterization as $\mathbf{\Pi}^0_2$ subspaces of countable powers of Sierpinski space, with several new proofs emphasizing this point of view as well as making more extensive use of Baire category techniques.