Abstract
Given uniformly homeomorphic metric spaces \(X\) and \(Y\), it is proven that 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 hyperspace \(C(X)\) is an Atsuji space 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 paper, we investigate space \(C(X)\) when \(X\) is an Atsuji space, and a class of Atsuji subspaces of \(C(X)\) is obtained. Using the results, some fixed point results for continuous maps on Atsuji spaces are obtained.
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Translated by E. Seifina
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Gupta, A.K., Mukherjee, S. Induced Homeomorphism and Atsuji Hyperspaces. Russ Math. 66, 8–15 (2022). https://doi.org/10.3103/S1066369X22100061
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DOI: https://doi.org/10.3103/S1066369X22100061