A note on KC Wallman compactifications
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- by Darrell W. Hajek and Angel E. Jiménez PDF
- Proc. Amer. Math. Soc. 61 (1976), 176-178 Request permission
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
- Darrell W. Hajek, A characterization of $T_{3}$-spaces, Indiana Univ. Math. J. 23 (1973/74), 23–25. MR 326654, DOI 10.1512/iumj.1973.23.23003
- Darrell W. Hajek, Functions with continuous Wallman extensions, Czechoslovak Math. J. 24(99) (1974), 40–43. MR 365491
- Lynn A. Steen and J. Arthur Seebach Jr., Counterexamples in topology, Holt, Rinehart and Winston, Inc., New York-Montreal, Que.-London, 1970. MR 0266131
Additional Information
- © Copyright 1976 American Mathematical Society
- 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