## Not every minimal Hausdorff space is $e$-compact

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- by R. M. Stephenson PDF
- 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 $X$.## References

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## Additional Information

- © Copyright 1975 American Mathematical Society
- 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