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

HTML articles powered by AMS MathViewer

- by R. M. Stephenson
- Proc. Amer. Math. Soc.
**52**(1975), 381-389 - DOI: https://doi.org/10.1090/S0002-9939-1975-0423296-4
- PDF | 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

- Bernhard Banaschewski,
*Über Hausdorffsch-minimale Erweiterung von Räumen*, Arch. Math.**12**(1961), 355–365 (German). MR**142097**, DOI 10.1007/BF01650574 - M. P. Berri, J. R. Porter, and R. M. Stephenson Jr.,
*A survey of minimal topological spaces*, General Topology and its Relations to Modern Analysis and Algebra, III (Proc. Conf., Kanpur, 1968) Academia, Prague, 1971, pp. 93–114. MR**0278254** - Manuel P. Berri and R. H. Sorgenfrey,
*Minimal regular spaces*, Proc. Amer. Math. Soc.**14**(1963), 454–458. MR**152978**, DOI 10.1090/S0002-9939-1963-0152978-9 - Nicolas Bourbaki,
*Espaces minimaux et espaces complètement séparés*, C. R. Acad. Sci. Paris**212**(1941), 215–218 (French). MR**5322** - Stephen H. Hechler,
*On a notion of weak compactness in non-regular spaces*, Studies in topology (Proc. Conf., Univ. North Carolina, Charlotte, N.C., 1974; dedicated to Math. Sect. Polish Acad. Sci.), Academic Press, New York, 1975, pp. 215–237. MR**0358692** - Edwin Hewitt,
*On two problems of Urysohn*, Ann. of Math. (2)**47**(1946), 503–509. MR**17527**, DOI 10.2307/1969089 - F. Burton Jones,
*Moore spaces and uniform spaces*, Proc. Amer. Math. Soc.**9**(1958), 483–486. MR**93757**, DOI 10.1090/S0002-9939-1958-0093757-9 - F. Burton Jones,
*Hereditarily separable, non-completely regular spaces*, Topology Conference (Virginia Polytech. Inst. and State Univ., Blacksburg, Va., 1973) Lecture Notes in Math., Vol. 375, Springer, Berlin, 1974, pp. 149–152. MR**0413044**
. M. Katětov, - R. M. Stephenson Jr.,
*Minimal first countable Hausdorff spaces*, Pacific J. Math.**36**(1971), 819–825. MR**288720**, DOI 10.2140/pjm.1971.36.819 - R. M. Stephenson,
*Two $R$-closed spaces*, Canadian J. Math.**24**(1972), 286–292. MR**298613**, DOI 10.4153/CJM-1972-023-5 - A. Tychonoff,
*Über die topologische Erweiterung von Räumen*, Math. Ann.**102**(1930), no. 1, 544–561 (German). MR**1512595**, DOI 10.1007/BF01782364

*Über*$H$-

*abgeschlossen und bikompakt Räume*, Časopis, Pěst. Mat. Fys.

**69**(1940), 36-49. MR

**1**, 317.

## Bibliographic 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