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Some compactness properties related to pseudocompactness and ultrafilter convergence. (English) Zbl 1266.54008

The author generalizes some notions of compactness and convergence with respect to an ultrafilter. For instance, \(\mathcal{F}-D\)-compactness relative to some family \(\mathcal{F}\) where \(D\) is an ultrafilter on a set \(Z\) and \(\mathcal{F}\) is a family of subset of a space \(X\). This notion is studied when \(\mathcal{F}\) is either the family of all singletons of \(X\) or the family of nonempty open subsets of \(X\). The concept of accumulation points is also generalized and the notion of \(\mathcal{F}-CAP_\lambda\) is introduced. The standard \([\mu,\lambda]\)-compactness is extended in two directions: one is named \(\mathcal{F}-[\mu,\lambda]\)-compactness and the other is called \([\mu,\lambda]\)-compactness relative to \(\mathcal{F}\). In the paper, the behavior of the products of this kind of spaces is studied. Several characterizations of these concepts are proved.

MSC:

54A20 Convergence in general topology (sequences, filters, limits, convergence spaces, nets, etc.)
54D20 Noncompact covering properties (paracompact, Lindelöf, etc.)
54B10 Product spaces in general topology
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