Following Kombarov we say that $X$ is $p$-sequential, for $p\in\alpha^\ast$, if for every non-closed subset $A$ of $X$ there is $f\in{}^\alpha X$ such that $f(\alpha)\subseteq A$ and $\bar f(p)\in X\backslash A$. This suggests the following definition due to Comfort and Savchenko, independently: $X$ is a {\rm FU($p$)}-space if for every $A\subseteq X$ and every $x\in A^{-}$ there is a function $f\in {}^\alpha A$ such that $\bar f(p)=x$. It is not hard to see that $p \leq {\,_{\operatorname{RK}}} q$ ($\leq {\,_{\operatorname{RK}}}$ denotes the Rudin--Keisler order) $\Leftrightarrow $ every $p$-sequential space is $q$-sequential $\Leftrightarrow $ every {\rm FU($p$)}-space is a {\rm FU($q$)}-space. We generalize the spaces $S_n$ to construct examples of $p$-sequential (for $p\in U(\alpha )$) spaces which are not {\rm FU($p$)}-spaces. We slightly improve a result of Boldjiev and Malykhin by proving that every $p$-sequential (Tychonoff) space is a {\rm FU($q$)}-space $\Leftrightarrow \forall \nu <\omega _1 (p^\nu \leq {\,_{\operatorname{RK}}} q)$, for $p,q \in \omega ^\ast $; and $S_n$ is a {\rm FU($p$)}-space for $p\in \omega ^\ast $ and $1
@article{116952,
author = {Salvador Garc\'\i a-Ferreira},
title = {On FU($p$)-spaces and $p$-sequential spaces},
journal = {Commentationes Mathematicae Universitatis Carolinae},
volume = {32},
year = {1991},
pages = {161-171},
zbl = {0789.54032},
mrnumber = {1118299},
language = {en},
url = {http://dml.mathdoc.fr/item/116952}
}
García-Ferreira, Salvador. On FU($p$)-spaces and $p$-sequential spaces. Commentationes Mathematicae Universitatis Carolinae, Tome 32 (1991) pp. 161-171. http://gdmltest.u-ga.fr/item/116952/
Martin's axiom and the construction of homogeneous bicompacta of countable tightness, Soviet Math. Dokl. 17 (1976), 256-260. (1976)
Ordinal invariants for topological spaces, Michigan Math. J. 15 (1968), 313-320. (1968) | MR 0240767
On compact Hausdorff spaces of countable tightness, Proc. Amer. Math. Soc. 105 (1989), 755-764. (1989) | MR 0930252 | Zbl 0687.54006
A new kind of compactness for topological spaces, Fund. Math. 66 (1970), 185-193. (1970) | MR 0251697 | Zbl 0198.55401
The sequentiality is equivalent to the $\Cal F$-Fréchet-Urysohn property, Comment. Math. Univ. Carolinae 31 (1990), 23-25. (1990) | MR 1056166 | Zbl 0696.54020
Ultrafilters on a countable set, Ann. Math. Logic 2 (1970), 1-24. (1970) | MR 0277371 | Zbl 0231.02067
Ultrafilters: some old and some new results, Bull. Amer. Math. Soc. 83 (1977), 417-455. (1977) | MR 0454893
On families of large oscillation, Fund. Math. 75 (1972), 275-290. (1972) | MR 0305343 | Zbl 0235.54005
The Theory of Ultrafilters, Grundlehren der Mathematischen Wissenschaften Vol. 211, Springer-Verlag, 1974. | MR 0396267 | Zbl 0298.02004
Fully closed mappings and the compatibility of some theorems of general topology with the axioms of set-theory, Math. USSR Sbornik 28 (1976), 1-26. (1976)
Various Orderings on the Space of Ultrafilters, Doctoral Dissertation, Wesleyan University, 1990.
Three Orderings on $\beta (ømega)\setminus ømega $, preprint. | MR 1227550 | Zbl 0791.54032
On a theorem of A. H. Stone, Soviet Math. Dokl. 27 (1983), 544-547. (1983) | Zbl 0531.54007
Compactness and sequentiality with respect to a set of ultrafilters, Moscow Univ. Math. Bull. 40 (1985), 15-18. (1985) | MR 0814266 | Zbl 0602.54025
An easier proof of the Shelah $P$-point independence theorem, Rapport 78, Wiskundig Seminarium, Free University of Amsterdam.
Convergence with respect to ultrafilters and the collective normality of products, Moscow Univ. Math. Bull. 43 (1988), 45-47. (1988) | MR 0938072 | Zbl 0687.54004
The Shelah $P$-point independence theorem, Israel J. Math. 43 (1982), 28-48. (1982) | MR 0728877 | Zbl 0511.03022