Selections and deleted symmetric products (Q1674141)

For a Hausdorff space \(X\) and a positive integer \(n\), let \([X]^n\) be the \(n\)-fold deleted symmetric product, that is, the hyperspace \(\{ S \subset X: |S|=n\}\) with the Vietoris topology, where \(|S|\) denotes the cardinality of \(S\). A map \(f: [X]^n \to X\) is called a selection for \([X]^n\) if \(f(S) \in S\) for every \(S \in [X]^n\). A selection for \([X]^2\) is called a weak selection for \(X\). \textit{M. Hrušák} and \textit{I. Martínez-Ruiz} [Fundam. Math. 203, No. 1, 1--20 (2009; Zbl 1170.54008)] proved that there exists a separable space \(X\) which admits a continuous selection for \([X]^3\) and has no continuous weak selection, and asked whether there exists a second countable space \(X\) that admits a continuous weak selection for \([X]^n\) for some \(n>2\), but does not admit a continuous weak selection. In this paper, the authors answer the above question affirmatively by giving, for any \(n\geq 2\), an example of a connected second countable space \(X\) such that \([X]^{n+1}\) admits a continuous selection, but \([X]^n\) has no continuous selection. The example is given by means of the topologist's sine curve. Furthermore, the authors prove that for a connected space \(X\) and \(n\geq 2\), \([X]^n\) has a continuous selection if and only if \([X]^{n+1}\) has at least two continuous selections. Orderability and weak orderability of a connected space with a continuous weak selection for \([X]^n\) for some \(n\geq 2\) are also discussed.