Hyperspaces with the Hausdorff Metric and Uniform ANR's

Abstract

For a metric space $X=(X,d)$,let ${\mathrm{C}\mathrm{l}\mathrm{d}}_H\left(X\right)$ be the space of all nonempty closed sets in $X$ with the topology induced by the Hausdorff extended metric:$$d_H(A,B)=\max \left\{\underset{x\in B}{\sup }d(x,A),\underset{x\in A}{\sup }d(x,B)\right\}\in [0,\mathrm{\infty }].$$ On each component of ${\mathrm{C}\mathrm{l}\mathrm{d}}_H\left(X\right)$, $d_H$ is a metric (i.e., ). In this paper, we give a condition on $X$ such that each component of ${\mathrm{C}\mathrm{l}\mathrm{d}}_H\left(X\right)$ is a uniform AR (in the sense of E. Michael). For a totally bounded metric space $X$, in order that ${\mathrm{C}\mathrm{l}\mathrm{d}}_H\left(X\right)$ is a uniform ANR,a necessary and sufficient condition is also given. Moreover, we discuss the subspace ${\mathrm{D}\mathrm{i}\mathrm{s}}_H\left(X\right)$ of ${\mathrm{C}\mathrm{l}\mathrm{d}}_H\left(X\right)$ consisting of all discrete sets in $X$ and give a condition on $X$ such that each component of ${\mathrm{D}\mathrm{i}\mathrm{s}}_H\left(X\right)$ is a uniform AR and ${\mathrm{D}\mathrm{i}\mathrm{s}}_H\left(X\right)$ is homotopy dense in ${\mathrm{C}\mathrm{l}\mathrm{d}}_H\left(X\right)$.