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On \(FU(p)\)-spaces and \(p\)-sequential spaces. (English) Zbl 0789.54032
The author investigates some natural generalizations of some topological properties defined in terms of \(\alpha\)-sequences and their \(p\)-limits, where \(p\) is an ultrafilter on the infinite cardinal \(\alpha\). Using Rudin-Keisler order of ultrafilters, several relationships between topological spaces are described, e.g. “every \(p\)-sequential space is \(q\)-sequential”, “every \(FU(p)\)-space is an \(FU(q)\)-space”, “every \(p\)-sequential space is an \(FU(q)\)-space.
The sequential spaces \(S_ n\) of A. V. Arkhangel’skiĭ and S. P. Franklin [Mich. Math. J. 15, 313–320 (1968; Zbl 0167.51102)] are generalized and studied.
The following result is also proved: if \(X\) is a space of tightness not greater than \(\alpha\) and \(| X | \leq 2^ \alpha\), then there is a uniform ultrafilter \(p\) on \(\alpha\) such that \(X\) is an \(FU(p)\)-space.

MSC:
54D55 Sequential spaces
54A25 Cardinality properties (cardinal functions and inequalities, discrete subsets)
03E05 Other combinatorial set theory
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