Isomorphisms of spaces of bounded continuous functions
Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part XI, Tome 113 (1981), pp. 243-246

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If $S$ is a locally compact metric space, countable at infinity, and the metric space $T$ is not locally compact, then the spaces $C(S)$ and $C(T)$ are not isomorphic.
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     author = {T. E. Khmyleva},
     title = {Isomorphisms of spaces of bounded continuous functions},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {243--246},
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     volume = {113},
     year = {1981},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_1981_113_a17/}
}
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T. E. Khmyleva. Isomorphisms of spaces of bounded continuous functions. Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part XI, Tome 113 (1981), pp. 243-246. http://geodesic.mathdoc.fr/item/ZNSL_1981_113_a17/