Selections and topologically well-ordered spaces (Q5946445)

Given a topological Hausdorff space \(X\), the Fell topology \(\tau_F\) on the hyperspace \(2^X\) of all nonempty closed subsets of \(X\) is the topology which has a subbase consisting of the sets of the form \(\{F\in 2^X: F\cap V\neq \emptyset\}\) and \(\{F\in 2^X : F\subseteq W\}\), where \(V\) and \(W\) are arbitrary nonempty open subsets of \(X\) with the restriction that the complement of \(W\) is compact (if this restriction is removed, we get the Vietoris topology \(\tau_V\)). \(X(2)\) stands for the hyperspace of all nonempty subsets of \(X\) consisting of at most two elements. A Fell continuous selector on a subspace \(\mathcal H\) of \(2^X\) is a \(\tau_F\)-continuous mapping \(\sigma:{\mathcal H}\to X\) such that \(\sigma(F)\in F\) for every \(f\in\mathcal H\). A topologically well-ordered space is a linearly ordered space in which every nonempty closed subset has a first element. The main result of the paper is the following equivalence.NEWLINENEWLINENEWLINETheorem. The following conditions are equivalent for a Hausdorff space \(X\): (i) there exists a Fell continuous selector on \(X(2)\); (ii) \(X\) is homeomorphic to a topologically well-ordered space. NEWLINENEWLINENEWLINEThe result extends the one due to \textit{V. Gutev} and \textit{T. Nogura} (preprint 1999) who proved that (ii) is equivalent to (iii) there exists a Fell continuous selector on \(2^X\).