Compactness and symmetric well-orders (Q6643763)

In this paper, the authors introduce and investigate a topological form of Stäckel's 1907 characterization of finite sets, aiming to develop an intriguing concept that characterizes usual compactness (or a close variant of it). A \(T_2\) topological space \((X, \tau)\) is defined to be Stäckel-compact if there exists a linear ordering \(\prec\) on \(X\) such that every non-empty \(\tau\)-closed set contains a \(\prec\)-least and a \(\prec\)-greatest element. The authors demonstrate that compact spaces are Stäckel-compact, but the converse does not hold. Additionally, Stäckel-compact spaces are shown to be countably compact. The equivalence of Stäckel-compactness with countable compactness remains an open problem. However, the main result establishes that this equivalence is valid in scattered spaces of Cantor-Bendixson rank \(< \omega_2\) under ZFC. Furthermore, under \(V = L\), the equivalence holds in all scattered spaces.