Developable hyperspaces are metrizable. (English) Zbl 1066.54009

Holá, L’ubica; Pelant, Jan; Zsilinszky, László

Appl. Gen. Topol. 4, No. 2, 351-360 (2003).

For a Hausdorff topological space \(X\) let \(CL(X)\) denote the hyperspace of all nonempty closed subsets of \(X\) and \(K(X)\) the hyperspace of all nonempty compact subsets of \(X\). In the context of \(CL(X)\) and \(K(X)\) endowed with various topologies (as Vietoris topology, Fell topology, locally finite topology, bounded Vietoris topology) the following properties are studied: developability, metrizability, having a \(G_{\delta}\)-diagonal, submetrizability. Some characterization theorems are proved. For example, the following are equivalent: (i) \(CL(X)\) with the Vietoris topology is Moore; (ii) \(CL(X)\) with the Vietoris topology is developable; (iii) \(CL(X)\) with the Vietoris topology is metrizable; (iv) \(X\) is compact and metrizable.

Reviewer: Władysław Makuchowski (Opole)

MSC:

54B20

Hyperspaces in general topology

Keywords:

developable spaces; metrizable space; Vietoris topology; Fell topology; locally finite topology; bounded Vietoris topology; \(G_{\delta}\)-diagonal