Induced Homeomorphism and Atsuji Hyperspaces

Abstract

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. 2020 Mathematics Subject Classification. 54B20.