Characterizing metric spaces whose hyperspaces are homeomorphic to $\ell _2$
Volume 113 / 2008
Colloquium Mathematicum 113 (2008), 223-229
MSC: 54B20, 57N20.
DOI: 10.4064/cm113-2-4
Abstract
It is shown that the hyperspace ${\rm Cld}_{\rm H}(X)$ (resp. ${\rm Bdd}_{\rm H}(X)$) of non-empty closed (resp. closed and bounded) subsets of a metric space $(X,d)$ is homeomorphic to $\ell_2$ if and only if the completion $\overline X$ of $X$ is connected and locally connected, $X$ is topologically complete and nowhere locally compact, and each subset (resp. each bounded subset) of $X$ is totally bounded.
