A note on KC Wallman compactifications

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.