A Few Projective Classes of (Non-Hausdorff) Topological Spaces

TL;DR

This paper investigates the projective properties of various classes of non-Hausdorff topological spaces, identifying which classes are projective or $ ext{omega}$-projective based on their properties.

Contribution

It extends the understanding of projective classes to non-Hausdorff spaces, characterizing classes like sober, compact sober, and strongly sober spaces as projective.

Findings

Sober and compact sober spaces are projective.

Most locally compact classes are not $ ext{omega}$-projective.

Certain classes like coherent sober and strongly sober spaces are projective.

Abstract

A class of topological spaces is projective (resp., $\omega$-projective) if and only if projective systems of spaces (resp., with a countable cofinal subset of indices) in the class are still in the class. A certain number of classes of Hausdorff spaces are known to be, or not to be, ($\omega$-) projective. We examine classes of spaces that are not necessarily Hausdorff. Sober and compact sober spaces form projective classes, but most classes of locally compact spaces are not even $\omega$-projective. Guided by the fact that the stably compact spaces are exactly the locally compact, strongly sober spaces, and that the strongly sober spaces are exactly the sober, coherent, compact, weakly Hausdorff (in the sense of Keimel and Lawson) spaces, we examine which classes defined by combinations of those properties are projective. Notably, we find that coherent sober spaces, compact coherent sober spaces, as well as (locally) strongly sober spaces, form projective classes.