Compactness and Symmetric Well Orders

Abstract

We introduce and investigate a topological version of Stäckel's 1907 characterization of finite sets, with the goal of obtaining an interesting notion that characterizes usual compactness (or a close variant of it). Define a $T_2$ topological space $(X,\tau )$ to be Stäckel-compact if there is some linear ordering $\prec$ on $X$ such that every non-empty $\tau$-closed set contains a $\prec$-least and a $\prec$-greatest element. We find that compact spaces are Stäckel-compact but not conversely, and Stäckel-compact spaces are countably compact. The equivalence of Stäckel-compactness with countable compactness remains open, but our main result is that this equivalence holds in scattered spaces of Cantor-Bendixson rank $<{\omega }_2$ under ZFC. Under V=L, the equivalence holds in all scattered spaces.

This is the accepted manuscript (with minor typo fixes) for the article published online on 17 Jan 2024 in Bulletin Polish Acad. Sci. Math. The journal allows free downloading of such Online First versions from the journal website under CC-BY license, and posting it on archives like arxiv.org. The published version's title is slightly different ("well orders" is hyphenated as "well-orders")