Induced homeomorphism and Atsuji hyperspaces

A. K. Gupta ; S. Mukherjee

Izvestiâ vysših učebnyh zavedenij. Matematika, no. 10 (2022), pp. 11-21.

Résumé

Given uniformly homeomorphic metric spaces $X$ and $Y$, it is proved that the hyperspaces $C(X)$ and $C(Y)$ are uniformly homeomorphic, where $C(X)$ denotes the collection of all nonempty closed subsets of $X$, and is endowed with Hausdorff distance. Gerald Beer has proved that the hyperspace $C(X)$ is Atsuji when $X$ is either compact or uniformly discrete. An Atsuji space is a generalization of compact metric spaces as well as of uniformly discrete spaces. In this article, we investigate the space $C(X)$ when $X$ is Atsuji, and a class of Atsuji subspaces of $C(X)$ is obtained. Using the obtained results, some fixed point results for continuous maps on Atsuji spaces are obtained.

Détail

Keywords: metric space, homeomorphism, Atsuji space, multivalued map.
Mots-clés : Hausdorff distance

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