## 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:$${d}_{H}(A,B)=\mathrm{max}\left\{\underset{x\in B}{\mathrm{sup}}d(x,A),\underset{x\in A}{\mathrm{sup}}d(x,B)\right\}\in [0,\mathrm{\infty}].$$ 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 ${\mathrm{C}\mathrm{l}\mathrm{d}}_{H}\left(X\right)$.

## Citation

Masayuki KURIHARA. Katsuro SAKAI. Masato YAGUCHI. "Hyperspaces with the Hausdorff Metric and Uniform ANR's." J. Math. Soc. Japan 57 (2) 523 - 535, April, 2005. https://doi.org/10.2969/jmsj/1158242069

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