Hyperspaces of separable Banach spaces with the Wijsman topology
Let \(\text{Cld}(X)\) be the set of all non-empty closed sets in a topological space \(X\). Let \(X= (X,d)\) be a metric space, for each \(x\in X\) and \(r> 0\), let \[ U^-(x,r)= \{A\in \text{Cld}(X): d(x,A)< r\};\;U^+(x,r)= \{A\in\text{Cld}(X): d(x,A)> r\}. \] The Wijsman topology on \(\text{Cld}(X)\) is the topology induced by the family \(\{U^-(x,r), U^+(x,r): x\in X, r> 0\}\) as an open subbasis and the space \(\text{Cld}(X)\) with the Wijsman topology is denoted by \(\text{Cld}_W(X)\). The authors discuss \(\text{Cld}_W(X)\) for separable metric spaces relating to Banach space and retract theory, and proved the following theorems. Theorem I. If \(X\) is an infinite-dimensional separable Banach space, then \(\text{Cld}_W(X)\) is homeomorphic to the separable Hilbert space \(\ell_2\). Theorem II. If \(x\) is an infinite-dimensional separable Banach space, then \(\text{Fin}_W(X)\!\approx\! \text{Bdd}_W(X)\!\approx \ell_2\times \ell^f_2\), where \(\text{Fin}_W(X)\) and \(\text{Bdd}_W(X)\) are the subspaces of \(\text{Cld}_W(X)\) consisting of all non-empty finite sets and consisting of all non-empty bounded closed sets, respectively, and \[ \ell^f_2= \{(x_i)_{i\in\mathbb{N}}\in \ell_2:x_i= 0\text{ except for finitely many }i\in\mathbb{N}\}. \] Theorem III. Let \(X\) be a separable metric space. If the component of any finite union of open balls in \(X\) has only finitely many path-components and all of them are unbounded closed (respectively each of them is compact or unbounded closed), then \(\text{Cld}_W(X)\) is an AR (respectively an ANR) and \(\text{Fin}_W(X)\) is homotopy dense in \(\text{Cld}_W(X)\). In particular, \({\mathfrak h}\) is an AR (respectively an ANR) if \(\text{Fin}_W(X)\subset{\mathfrak h}\subset \text{Cld}_W(X)\). It is said that \(Y\subset X\) is homotopy dense in \(X\) if there is a homotopy \(h: X\times 1\to X\) such that \(h_0= \text{id}\) and \(h_t(X)\in Y\) for every \(t> 0\). Theorem IV. Let \(X\) be a separable metric space. If the component of any finite union of open balls in \(X\) has only finitely many path-components, all of which are closed in \(X\), then \(\text{Fin}_W(X)\) and \(\text{Cld}_W(X)\) are ANRs. Further, the authors give an example such that \(Y\) is a closed subset of \(X\) and the Wijsman topology on \(\text{Cld}(Y)\) does not coincide with the subspace topology of \(\text{Cld}_W(X)\) on \(\text{Cld}(Y)\); they also discuss \(\text{Cld}(Y)\) with the subspace topology inherited from \(\text{Cld}_W(X)\), which is called the relative Wijsman topology.
infinite-dimensional separable Banach space
scientific article; zbMATH DE number 2142277
scientific article