Compactness and symmetric well-orders

Abhijit Dasgupta Bulletin Polish Acad. Sci. Math. MSC: Primary 54D30; Secondary 03E35, 54F05, 54G12 DOI: 10.4064/ba230424-28-12 Published online: 17 January 2024


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 $T_2$ 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 $ \lt \omega _2$ under ZFC. Under $V=L$, the equivalence holds in all scattered spaces.


  • Abhijit DasguptaDepartment of Mathematics
    University of Detroit Mercy
    Detroit, MI 48221, USA

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