Compactness and Symmetric Well Orders

TL;DR

This paper introduces a topological concept called St"ackel-compactness, exploring its relation to compactness and countable compactness, and establishes equivalences in certain classes of scattered spaces.

Contribution

It defines St"ackel-compactness as a new topological property and proves its equivalence to countable compactness in specific classes of scattered spaces.

Findings

Compact spaces are St"ackel-compact but not vice versa

St"ackel-compact spaces are countably compact

Equivalence of St"ackel-compactness and countable compactness holds in certain scattered spaces

Abstract

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.