Some compactness properties related to pseudocompactness and ultrafilter convergence
Abstract
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 .
Cite
@article{arxiv.0907.0602,
title = {Some compactness properties related to pseudocompactness and ultrafilter convergence},
author = {Paolo Lipparini},
journal= {arXiv preprint arXiv:0907.0602},
year = {2011}
}
Comments
v2, largely expanded and entirely rewritten; 19 pages v3 corrected an inaccuracy (see Example 3.4) and expanded Section 6; 25 pages