  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.
