Not every minimal Hausdorff space is 𝑒-compact

Stephenson, R. M.

doi:10.1090/S0002-9939-1975-0423296-4

Not every minimal Hausdorff space is $e$-compact
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Proc. Amer. Math. Soc. 52 (1975), 381-389 Request permission

Abstract:

A topological space $X$ is said to be $e$-compact with respect to a dense subset $D$ provided either of the following equivalent conditions is satisfied: (i) every open cover of $X$ has a finite subcollection which covers $D$; (ii) every ultrafilter on $D$ converges to a point of $X$. If there exists a dense subset with respect to which a space $X$ is $e$-compact, then $X$ is called $e$-compact.$^{1}$ Two problems recently raised by S. H. Hechler are the following. (a) Is every minimal Hausdorff space $e$-compact? (b) If there exists a Hausdorff space which is $e$-compact with respect to a space $D$, must $D$ be completely regular? The main purpose of this paper is to provide a negative answer to (a) and to present some results which the author hopes will be of use in the solution to (b). These results can also be used to obtain a construction of $\beta X$ for certain completely regular Hausdorff spaces $X$.

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