Continuous selections and \(C\)-spaces (Q2750910)

A space \(X\) is said to be a \(C\)-space provided that for any sequence \(\{\omega_n:n \in\mathbb{N}\}\) of open covers of \(X\) there exists a sequence \(\{\gamma_n: n\in\mathbb{N}\}\) of open disjoint families in \(X\) such that \(\gamma_n\) refines \(\omega_n\) and \(\bigcup \{\gamma_n:n \in\mathbb{N}\}\) is a cover of \(X\). The authors give characterizations of paracompact \(C\)-spaces via continuous selections avoiding \(Z_\infty\)-set. The results are applied to prove a countable sum theorem for paracompact \(C\)-spaces, and to obtain the following theorem being a partial answer to a question of E. Michael. Let \(X\) be a paracompact \(C\)-space, \(E\) be a Banach space, \(Y\) be a \(G_\delta\)-subset of \(E\), and let \({\mathcal F}_c (Y)\) denote the family of all convex closed nonempty subsets of \(Y\). Then every lower semi-continuous mapping \(\varphi:X \to{\mathcal F}_c(Y)\) admits a continuous selection.