Geodesic
Induced homeomorphism and Atsuji hyperspaces
Izvestiâ vysših učebnyh zavedenij. Matematika, no. 10 (2022), pp. 11-21

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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.
Keywords: metric space, homeomorphism, Atsuji space, multivalued map.
Mots-clés : Hausdorff distance 
@article{IVM_2022_10_a1,
     author = {A. K. Gupta and S. Mukherjee},
     title = {Induced homeomorphism and {Atsuji} hyperspaces},
     journal = {Izvesti\^a vys\v{s}ih u\v{c}ebnyh zavedenij. Matematika},
     pages = {11--21},
     publisher = {mathdoc},
     number = {10},
     year = {2022},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/IVM_2022_10_a1/}
}
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A. K. Gupta; S. Mukherjee. Induced homeomorphism and Atsuji hyperspaces. Izvestiâ vysših učebnyh zavedenij. Matematika, no. 10 (2022), pp. 11-21. http://geodesic.mathdoc.fr/item/IVM_2022_10_a1/