Induced Homeomorphism and Atsuji Hyperspaces

TL;DR

This paper explores the properties of hyperspaces derived from Atsuji metric spaces, showing they are uniformly homeomorphic under certain conditions and establishing fixed point results for continuous maps.

Contribution

It demonstrates that hyperspaces of Atsuji spaces retain Atsuji properties and introduces a class of Atsuji subspaces, extending understanding of their structure.

Findings

Hyperspaces of Atsuji spaces are uniformly homeomorphic under uniform homeomorphisms.

A class of Atsuji subspaces within hyperspaces is identified.

Fixed point results are established for continuous maps on Atsuji spaces.

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.