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On the completeness of the space of compact subsets


Author: Phillip Zenor
Journal: Proc. Amer. Math. Soc. 26 (1970), 190-192
MSC: Primary 54.20
DOI: https://doi.org/10.1090/S0002-9939-1970-0261538-4
MathSciNet review: 0261538
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Abstract: The purpose of this paper is to show that if $ X$ is a completely regular $ {T_1}$-space, then $ \mathcal{C}(X)$, the space of all compact subsets of $ X$ with the Vietoris topology, is realcompact (topologically complete in the sense of Dieudonné) if and only if $ X$ is realcompact (topologically complete in the sense of Dieudonné).


References [Enhancements On Off] (What's this?)

  • [1] E. Michael, Topologies on spaces of subsets, Trans. Amer. Math. Soc. 71 (1951), 152-182. MR 13, 54. MR 0042109 (13:54f)
  • [2] P. Zenor, Extending completely regular spaces with inverse limits, Glasnik Mat. Ser III 5 (1970). MR 0275371 (43:1128)
  • [3] -, Extensions of topological spaces, (to appear).

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1970-0261538-4
Keywords: Hyperspace, realcompactness, completeness, inverse limits
Article copyright: © Copyright 1970 American Mathematical Society

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