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

García-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 -