Compactness and Symmetric Well Orders

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.