Some compactness properties related to pseudocompactness and ultrafilter convergence

Details Some compactness properties related to pseudocompactness and ultrafilter convergence Some compactness properties related to pseudocompactness and ultrafilter convergence

2009-07-03

arxiv.org/abs/0907.0602v3

Topology Proceedings, Volume 40 (2012) Pages 29-51

Mathematics

General Topology

v2, largely expanded and entirely rewritten; 19 pages v3 corrected an inaccuracy (see Example 3.4) and expanded Section 6; 25

Scientific paper

We discuss some notions of compactness and convergence relative to a specified family F of subsets of some topological space X. The two most interesting particular cases of our construction appear to be the following ones. (1) The case in which F is the family of all singletons of X, in which case we get back the more usual notions. (2) The case in which F is the family of all nonempty open subsets of X, in which case we get notions related to pseudocompactness. A large part of the results in this note are known in particular case (1); the results are, in general, new in case (2). As an example, we characterize those spaces which are D-pseudocompact, for some ultrafilter D uniform over $\lambda$.

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