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

Abstract

A class of topological spaces is projective (resp., $ω$-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, ($ω$-) 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 $ω$-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.