Characterizations of compact ordinal spaces via continuous selections (Q1295179)

Let \(X\) be a compact Hausdorff space. It is proved, that \(X\) is homeomorphic to an ordinal space if and only if there exists a continuous selection \(f:{\mathcal F}(X)\to X\) on the hyperspace \({\mathcal F}(X)\) of closed nonempty subsets of \(X\) such that \(f(F)\) is an isolated point of \(F\) for every \(F\in {\mathcal F}(X)\). The paper contains also examples and a problem concerning the characterization of spaces of ordinals via continuous selections under the condition of local compactness.