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Proceedings of the American Mathematical Society

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

 
 

 

Not every minimal Hausdorff space is $e$-compact


Author: R. M. Stephenson
Journal: Proc. Amer. Math. Soc. 52 (1975), 381-389
MSC: Primary 54D25
DOI: https://doi.org/10.1090/S0002-9939-1975-0423296-4
MathSciNet review: 0423296
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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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Keywords: Stone-&#268;ech compactifications, Banaschewski minimal Hausdorff completions, absolutely closed spaces, <IMG WIDTH="15" HEIGHT="20" ALIGN="BOTTOM" BORDER="0" SRC="images/img3.gif" ALT="$e$">-compact spaces
Article copyright: © Copyright 1975 American Mathematical Society