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 $S$ 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.
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J. Ginsburg and V. Saks, Some applications of ultrafilters in topology, Pacific J. Math. 57 (1975), 403-418.
I. Juhász, Cardinal functions in topology, Math. Centre Tracts, no. 34, Math. Centrum, Amsterdam, 1971.