# On FU($p$)-spaces and $p$-sequential spaces

Commentationes Mathematicae Universitatis Carolinae (1991)

- Volume: 32, Issue: 1, page 161-171
- ISSN: 0010-2628

## Access Full Article

top## Abstract

top## How to cite

topGarcía-Ferreira, Salvador. "On FU($p$)-spaces and $p$-sequential spaces." Commentationes Mathematicae Universitatis Carolinae 32.1 (1991): 161-171. <http://eudml.org/doc/247273>.

@article{García1991,

abstract = {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 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 $p \le \{\,_\{\operatorname\{RK\}\}\} q$ ($\le \{\,_\{\operatorname\{RK\}\}\}$ denotes the Rudin–Keisler order) $\Leftrightarrow $ every $p$-sequential space is $q$-sequential $\Leftrightarrow $ every FU($p$)-space is a FU($q$)-space. We generalize the spaces $S_n$ to construct examples of $p$-sequential (for $p\in U(\alpha )$) spaces which are not FU($p$)-spaces. We slightly improve a result of Boldjiev and Malykhin by proving that every $p$-sequential (Tychonoff) space is a FU($q$)-space $\Leftrightarrow \forall \nu <\omega _1 (p^\nu \le \{\,_\{\operatorname\{RK\}\}\} q)$, for $p,q \in \omega ^\ast $; and $S_n$ is a FU($p$)-space for $p\in \omega ^\ast $ and $1<n<\omega \Leftrightarrow $ every sequential space $X$ with $\sigma (X)\le n$ is a FU($p$)-space $\Leftrightarrow \exists \lbrace p_\{n-2\}, \dots , p_1\rbrace \subseteq \omega ^\ast (p_\{n-2\}<\{\,_\{\operatorname\{RK\}\}\} \dots <\{\,_\{\operatorname\{RK\}\}\} p_1 <_\{\,l\} p)$; hence, it is independent with ZFC that $S_3$ is a FU($p$)-space for all $p\in \omega ^\ast $. It is also shown that $|\beta (\alpha )\setminus U(\alpha )|\le 2^\alpha \Leftrightarrow $ every space $X$ with $t(X)<\alpha $ is $p$-sequential for some $p\in U(\alpha ) \Leftrightarrow $ every space $X$ with $t(X)<\alpha $ is a FU($p$)-space for some $p\in U(\alpha )$; if $t(X)\le \alpha $ and $|X|\le 2^\alpha $, then $ \exists p\in U(\alpha ) $ ($X$ is a FU($p$)-space).},

author = {García-Ferreira, Salvador},

journal = {Commentationes Mathematicae Universitatis Carolinae},

keywords = {ultrafilter; Rudin–Frolík order; Rudin–Keisler order; $p$-compact; quasi $M$-compact; strongly $M$-sequential; weakly $M$-sequential; $p$-sequential; FU($p$)-space; sequential; $P$-point; -sequential space; Rudin-Frolík order; -space; Rudin- Keisler order; ultrafilter},

language = {eng},

number = {1},

pages = {161-171},

publisher = {Charles University in Prague, Faculty of Mathematics and Physics},

title = {On FU($p$)-spaces and $p$-sequential spaces},

url = {http://eudml.org/doc/247273},

volume = {32},

year = {1991},

}

TY - JOUR

AU - García-Ferreira, Salvador

TI - On FU($p$)-spaces and $p$-sequential spaces

JO - Commentationes Mathematicae Universitatis Carolinae

PY - 1991

PB - Charles University in Prague, Faculty of Mathematics and Physics

VL - 32

IS - 1

SP - 161

EP - 171

AB - 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 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 $p \le {\,_{\operatorname{RK}}} q$ ($\le {\,_{\operatorname{RK}}}$ denotes the Rudin–Keisler order) $\Leftrightarrow $ every $p$-sequential space is $q$-sequential $\Leftrightarrow $ every FU($p$)-space is a FU($q$)-space. We generalize the spaces $S_n$ to construct examples of $p$-sequential (for $p\in U(\alpha )$) spaces which are not FU($p$)-spaces. We slightly improve a result of Boldjiev and Malykhin by proving that every $p$-sequential (Tychonoff) space is a FU($q$)-space $\Leftrightarrow \forall \nu <\omega _1 (p^\nu \le {\,_{\operatorname{RK}}} q)$, for $p,q \in \omega ^\ast $; and $S_n$ is a FU($p$)-space for $p\in \omega ^\ast $ and $1<n<\omega \Leftrightarrow $ every sequential space $X$ with $\sigma (X)\le n$ is a FU($p$)-space $\Leftrightarrow \exists \lbrace p_{n-2}, \dots , p_1\rbrace \subseteq \omega ^\ast (p_{n-2}<{\,_{\operatorname{RK}}} \dots <{\,_{\operatorname{RK}}} p_1 <_{\,l} p)$; hence, it is independent with ZFC that $S_3$ is a FU($p$)-space for all $p\in \omega ^\ast $. It is also shown that $|\beta (\alpha )\setminus U(\alpha )|\le 2^\alpha \Leftrightarrow $ every space $X$ with $t(X)<\alpha $ is $p$-sequential for some $p\in U(\alpha ) \Leftrightarrow $ every space $X$ with $t(X)<\alpha $ is a FU($p$)-space for some $p\in U(\alpha )$; if $t(X)\le \alpha $ and $|X|\le 2^\alpha $, then $ \exists p\in U(\alpha ) $ ($X$ is a FU($p$)-space).

LA - eng

KW - ultrafilter; Rudin–Frolík order; Rudin–Keisler order; $p$-compact; quasi $M$-compact; strongly $M$-sequential; weakly $M$-sequential; $p$-sequential; FU($p$)-space; sequential; $P$-point; -sequential space; Rudin-Frolík order; -space; Rudin- Keisler order; ultrafilter

UR - http://eudml.org/doc/247273

ER -

## References

top- Arhangel'skii A.V., Martin's axiom and the construction of homogeneous bicompacta of countable tightness, Soviet Math. Dokl. 17 (1976), 256-260. (1976)
- Arhangel'skii A.V., Franklin S.P., Ordinal invariants for topological spaces, Michigan Math. J. 15 (1968), 313-320. (1968) MR0240767
- Balogh Z., On compact Hausdorff spaces of countable tightness, Proc. Amer. Math. Soc. 105 (1989), 755-764. (1989) Zbl0687.54006MR0930252
- Bernstein A.R., A new kind of compactness for topological spaces, Fund. Math. 66 (1970), 185-193. (1970) Zbl0198.55401MR0251697
- Boldjiev B., Malykhin V., The sequentiality is equivalent to the $\mathcal{F}$-Fréchet-Urysohn property, Comment. Math. Univ. Carolinae 31 (1990), 23-25. (1990) Zbl0696.54020MR1056166
- Booth D.D., Ultrafilters on a countable set, Ann. Math. Logic 2 (1970), 1-24. (1970) Zbl0231.02067MR0277371
- Comfort W.W., Ultrafilters: some old and some new results, Bull. Amer. Math. Soc. 83 (1977), 417-455. (1977) MR0454893
- Comfort W.W., Negrepontis S., On families of large oscillation, Fund. Math. 75 (1972), 275-290. (1972) Zbl0235.54005MR0305343
- Comfort W.W., Negrepontis S., The Theory of Ultrafilters, Grundlehren der Mathematischen Wissenschaften Vol. 211, Springer-Verlag, 1974. Zbl0298.02004MR0396267
- Fedorčuk V.V., 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)
- Garcia-Ferreira S., Various Orderings on the Space of Ultrafilters, Doctoral Dissertation, Wesleyan University, 1990.
- Garcia-Ferreira S., Three Orderings on $\beta \left(\xf8mega\right)\setminus \xf8mega$, preprint. Zbl0791.54032MR1227550
- Kombarov A.P., On a theorem of A. H. Stone, Soviet Math. Dokl. 27 (1983), 544-547. (1983) Zbl0531.54007
- Kombarov A.P., Compactness and sequentiality with respect to a set of ultrafilters, Moscow Univ. Math. Bull. 40 (1985), 15-18. (1985) Zbl0602.54025MR0814266
- Mills Ch., An easier proof of the Shelah $P$-point independence theorem, Rapport 78, Wiskundig Seminarium, Free University of Amsterdam.
- Savchenko I.A., Convergence with respect to ultrafilters and the collective normality of products, Moscow Univ. Math. Bull. 43 (1988), 45-47. (1988) Zbl0687.54004MR0938072
- Wimmers E.L., The Shelah $P$-point independence theorem, Israel J. Math. 43 (1982), 28-48. (1982) Zbl0511.03022MR0728877

## Citations in EuDML Documents

top- Paolo Lipparini, A very general covering property
- Paolo Lipparini, Topological spaces compact with respect to a set of filters
- Salvador García-Ferreira, Quasi $M$-compact spaces
- Salvador García-Ferreira, Angel Tamariz-Mascarúa, On $p$-sequential $p$-compact spaces
- Salvador García-Ferreira, Paul J. Szeptycki, MAD families and $P$-points
- Salvador García-Ferreira, Angel Tamariz-Mascarúa, $p$-sequential like properties in function spaces

## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.