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

Garcia-Ferreira, Salvador

Commentationes Mathematicae Universitatis Carolinae, Tome 32 (1991) no. 1, pp. 161-171

Résumé

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 $1$ every sequential space $X$ with $\sigma (X)\leq n$ is a {\rm FU($p$)}-space $\Leftrightarrow \exists \{p_{n-2}, \dots , p_1\}\subseteq \omega ^\ast (p_{n-2}{\,_{\operatorname{RK}}} \dots {\,_{\operatorname{RK}}} p_1 $; hence, it is independent with ZFC that $S_3$ is a {\rm FU($p$)}-space for all $p\in \omega ^\ast $. It is also shown that $|\beta (\alpha )\setminus U(\alpha )|\leq 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 {\rm FU($p$)}-space for some $p\in U(\alpha )$; if $t(X)\leq \alpha $ and $|X|\leq 2^\alpha $, then $ \exists p\in U(\alpha ) $ ($X$ is a {\rm FU($p$)}-space).

Détail

Classification : 03E05, 04A20, 54A25, 54D55, 54D99
Keywords: ultrafilter; Rudin--Frol\'\i k order; Rudin--Keisler order; $p$-compact; quasi $M$-compact; strongly $M$-sequential; weakly $M$-sequential; $p$-sequential; FU($p$)-space; sequential; $P$-point