The hyperspace of the regions below continuous maps with the Fell topology
For a Tychonoff space \(X\) and \(\mathbf I=[0,1]\), \(\text{Cld}_F(X\times \mathbf I)\) stands for the hyperspace \(\text{Cld}(X\times \mathbf I)\) of all nonempty closed subsets of \(X\times \mathbf I\) endowed with the \textit{Fell topology} having as subbase sets of the form \(\{A\in \text{Cld}(X\times \mathbf I): A\cap U\neq\emptyset\}\), and \(\{A\in \text{Cld}(X\times \mathbf I): A\cap K=\emptyset\}\), where \(U\subseteq X\times \mathbf I\) is open and \(K\subseteq X\times \mathbf I\) is compact. If \(\downarrow USC(X)\), and \(\downarrow C(X)\) stand for the collection of hypographs \(\{(x,\lambda)\in X\times \mathbf I: \lambda \leq f(x)\}\) of upper semicontinuous and continuous maps \(f:X\to\mathbf I\), respectively, then \(\downarrow USC(X)\) and \(\downarrow C(X)\) can be viewed as subspaces of \(\text{Cld}_F(X\times \mathbf I)\). Motivated by analogous results for the so-called Vietoris hyperspace topology, the authors characterize under what conditions on \(X\) there is a homeomorphism \(h\) between \(\downarrow~USC(X)\) and \([-1,1]^{\omega}\) with \(h(\downarrow C(X))=c_0=\{(x_n)\in [-1,1]^{\omega}: \lim_n x_n=0\}\). The main result is that this happens if and only if \(X\) is locally compact, separable, metrizable, and the set of isolated points of \(X\) is not dense in \(X\).
scientific article; zbMATH DE number 6145089
scientific article