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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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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 $X$.
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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