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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
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
Summary:
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<n<\omega \Leftrightarrow $ every sequential space $X$ with $\sigma (X)\leq n$ is a {\rm FU($p$)}-space $\Leftrightarrow \exists \{p_{n-2}, \dots , p_1\}\subseteq \omega ^\ast (p_{n-2}<{\,_{\operatorname{RK}}} \dots <{\,_{\operatorname{RK}}} p_1 <_{\,l} p)$; 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).
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