Hyperspaces of normed linear spaces with the Attouch-Wets topology
For a metric space \((X,d)\) let Cld\((X)\) denote the space of all non-empty closed subsets of \(X\). The space Cld\((X)\) can be seen as a subset of \(C(X)\), the space of all continuous real valued functions, by identifying each set \(A\in \text{Cld}(X)\) with the function \(a\in C(X)\) defined by \(a(x)=d(x,A)\) for \(x\in X\). The Attouch-Wets Topology of Cld\((X)\) is inherited from \(C(X)\) with the topology of uniform convergence on bounded sets. The authors investigate some topological properties in the hyperspace Cld\((X)\) equipped with the Attouch-Wets topology. The main results say that for any normed linear space \(X\), the hyperspace Cld\((X)\) and various subspaces, including the following: Bdd\((X)\) of all bounded closed sets Reg\((X)\) of all regular (i.e. \(A=\text{cl(int}A)\)) sets Conn\((X)\) of all closed sets with finite components Comp\((X)\) of all compacta Cont\((X)\) of all finite unions of continua Pol\((X)\) of all compact polyhedra Fin\((X)\) of all finite sets are ARs. Moreover if \(X\) is an infinite-dimensional Banach space with weight \(w(X)\), then Cld\((X)\) is homeomorphic to the Hilbert space with weight \(2^{w(X)}\). In particular, if \(X\) is a separable infinite dimensional Banach space, then Cld\((X)\) is homeomorphic to the space \(m=\ell_\infty\).
scientific article; zbMATH DE number 1901529