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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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Spaces with dense conditionally compact subsets
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by Andrew J. Berner PDF
Proc. Amer. Math. Soc. 81 (1981), 137-142 Request permission

Abstract:

A subset $S$ of a topological space is said to be conditionally compact if every infinite subset of $S$ has a limit point in the space. If a space has a dense conditionally compact subset, it follows that it is pseudocompact, but the converse is not true. Examples are given of spaces that are pseudocompact, do not have dense conditionally compact subsets, but do have compactifications that are products of first countable spaces. For locally compact spaces, though, with such compactifications, the continuum hypothesis implies that pseudocompactness is equivalent to having a dense conditionally compact subset. A locally compact pseudocompact space without a dense conditionally compact subset is described.
References
  • John Ginsburg and Victor Saks, Some applications of ultrafilters in topology, Pacific J. Math. 57 (1975), no. 2, 403–418. MR 380736
  • I. Juhász, Cardinal functions in topology, Mathematical Centre Tracts, No. 34, Mathematisch Centrum, Amsterdam, 1971. In collaboration with A. Verbeek and N. S. Kroonenberg. MR 0340021
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Additional Information
  • © Copyright 1981 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 81 (1981), 137-142
  • MSC: Primary 54D30; Secondary 54D35, 54D45
  • DOI: https://doi.org/10.1090/S0002-9939-1981-0589156-6
  • MathSciNet review: 589156