Compactness and symmetric well-orders

We introduce and investigate a topological form 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 T2 topological space (X,τ) to be Stäckel-compact if there is some linear ordering ≺ on X such that every non-empty τ-closed set contains a ≺-least and a ≺-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 <ω2 under ZFC. Under V=L, the equivalence holds in all scattered spaces.