Spaces with dense conditionally compact subsets

Author:
Andrew J. Berner

Journal:
Proc. Amer. Math. Soc. **81** (1981), 137-142

MSC:
Primary 54D30; Secondary 54D35, 54D45

MathSciNet review:
589156

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Abstract: A subset of a topological space is said to be conditionally compact if every infinite subset of 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.

**[1]**John Ginsburg and Victor Saks,*Some applications of ultrafilters in topology*, Pacific J. Math.**57**(1975), no. 2, 403–418. MR**0380736****[2]**I. Juhász,*Cardinal functions in topology*, Mathematisch Centrum, Amsterdam, 1971. In collaboration with A. Verbeek and N. S. Kroonenberg; Mathematical Centre Tracts, No. 34. MR**0340021**

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DOI:
https://doi.org/10.1090/S0002-9939-1981-0589156-6

Keywords:
Pseudocompact,
conditionally compact

Article copyright:
© Copyright 1981
American Mathematical Society