Compactness and symmetric well-orders

Abhijit Dasgupta¹

¹ Department of Mathematics University of Detroit Mercy Detroit, MI 48221, USA

Bulletin of the Polish Academy of Sciences. Mathematics, Tome 72 (2024) no. 1, pp. 67-80.

Résumé

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. Published in Open Access (under CC-BY license).

Détail

DOI : 10.4064/ba230424-28-12

Keywords: introduce investigate topological form ckel characterization finite sets obtaining interesting notion characterizes usual compactness close variant define topological space tau ckel compact there linear ordering prec every non empty tau closed set contains prec least prec greatest element compact spaces ckel compact conversely ckel compact spaces countably compact equivalence ckel compactness countable compactness remains main result equivalence holds scattered spaces cantor bendixson rank omega under zfc under equivalence holds scattered spaces