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A note on KC Wallman compactifications


Authors: Darrell W. Hajek and Angel E. Jiménez
Journal: Proc. Amer. Math. Soc. 61 (1976), 176-178
MSC: Primary 54D35
DOI: https://doi.org/10.1090/S0002-9939-1976-0428283-9
MathSciNet review: 0428283
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Abstract: In a previous paper, D. W. Hajek showed that if a space $ X$ is a $ {T_3}$ space and $ A$ is a compact subset of $ WX$, the Wallman compactification of $ X$, then $ X \cap A$ is a closed subset of $ X$. This raises the question of whether this ``closed intersection'' property characterizes the $ {T_3}$ spaces among the Hausdorff spaces. In the present paper, the authors show this conjecture is false by giving an example of a nonregular Hausdorff space whose Wallman compactification is a $ \operatorname{KC} $ (compact closed)-space, and, hence, trivially satisfies this ``closed intersection'' property.


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

  • [1] D. Hajek, A characterization of $ {T_3}$ spaces, Indiana Univ. Math. J. 23 (1973/74), 23-25. MR 48 #4997. MR 0326654 (48:4997)
  • [2] -, Functions with continuous Wallman extensions, Czechoslovak Math. J. 24 (99) (1974), 40-43. MR 0365491 (51:1743)
  • [3] L. A. Steen and J. A. Seebach, Jr., Counterexamples in topology, Holt, Rinehart and Winston, New York, 1970. MR 42 # 1040. MR 0266131 (42:1040)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1976-0428283-9
Keywords: Wallman compactification, $ \operatorname{KC} $ (compact closed)-space, "closed intersection'' property, nonregular Hausdorff space
Article copyright: © Copyright 1976 American Mathematical Society

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