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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



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
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.

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Keywords: Wallman compactification, $ \operatorname{KC} $ (compact closed)-space, "closed intersection'' property, nonregular Hausdorff space
Article copyright: © Copyright 1976 American Mathematical Society