Fell-continuous selections and topologically well-orderable spaces. II (Q2782215)

For a space \(X\), let \({\mathcal F}(X)\) be its family of non-empty closed subsets. If \({\mathcal V}\) is a family of open subsets of \(X\) then \(\langle {\mathcal V}\rangle\) denotes the following collection: \(\{S\in {\mathcal F}(X) : (\forall V\in {\mathcal V})(S\cap V\not=\emptyset) \& (S\subset \bigcup {\mathcal V})\}\). A basis for the Fell topology on \({\mathcal F}(X)\) consists of all sets of the form \(\langle {\mathcal V}\rangle\) with the property that \(X\smallsetminus \bigcup {\mathcal V}\) is compact. A space \(X\) is topologically well-orderable if there is a linear order \(<\) on \(X\) which generates its topology while moreover every non-empty closed subset of \(X\) has a \(<\)-minimal element. Gutev and Nogura showed that a space \(X\) is topologically well-orderable if and only if \({\mathcal F}(X)\) has a selection which is continuous with respect to the Fell-topology on \({\mathcal F}(X)\). In this paper, the author extends this result in a nontrivial way: it suffices to assume the existence of a selection on \({\mathcal F}_2(X)\), the subspace of \({\mathcal F}(X)\) consisting of the at most 2-element subsets of \(X\).NEWLINENEWLINEFor the entire collection see [Zbl 0983.00046].