Compactness and Symmetric Well Orders

Abstract

We introduce and investigate a topological version of St\"ackel'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 St\"ackel-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 St\"ackel-compact but not conversely, and St\"ackel-compact spaces are countably compact. The equivalence of St\"ackel-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.