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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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