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

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

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## Abstract

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Following Kombarov we say that $X$ is $p$-sequential, for $p\in {\alpha }^{*}$, if for every non-closed subset $A$ of $X$ there is $f\in {}^{\alpha }X$ such that $f\left(\alpha \right)\subseteq A$ and $\overline{f}\left(p\right)\in X\setminus A$. 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 $\overline{f}\left(p\right)=x$. It is not hard to see that $p\le {\phantom{\rule{0.166667em}{0ex}}}_{RK}q$ ($\le {\phantom{\rule{0.166667em}{0ex}}}_{RK}$ denotes the Rudin–Keisler order) $⇔$ every $p$-sequential space is $q$-sequential $⇔$ every FU($p$)-space is a FU($q$)-space. We generalize the spaces ${S}_{n}$ to construct examples of $p$-sequential (for $p\in U\left(\alpha \right)$) 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 $⇔\forall \nu <{\omega }_{1}\left({p}^{\nu }\le {\phantom{\rule{0.166667em}{0ex}}}_{RK}q\right)$, for $p,q\in {\omega }^{*}$; and ${S}_{n}$ is a FU($p$)-space for $p\in {\omega }^{*}$ and $1 every sequential space $X$ with $\sigma \left(X\right)\le n$ is a FU($p$)-space $⇔\exists \left\{{p}_{n-2},\cdots ,{p}_{1}\right\}\subseteq {\omega }^{*}\left({p}_{n-2}<{\phantom{\rule{0.166667em}{0ex}}}_{RK}\cdots <{\phantom{\rule{0.166667em}{0ex}}}_{RK}{p}_{1}{<}_{\phantom{\rule{0.166667em}{0ex}}l}p\right)$; hence, it is independent with ZFC that ${S}_{3}$ is a FU($p$)-space for all $p\in {\omega }^{*}$. It is also shown that $|\beta \left(\alpha \right)\setminus U\left(\alpha \right)|\le {2}^{\alpha }⇔$ every space $X$ with $t\left(X\right)<\alpha$ is $p$-sequential for some $p\in U\left(\alpha \right)⇔$ every space $X$ with $t\left(X\right)<\alpha$ is a FU($p$)-space for some $p\in U\left(\alpha \right)$; if $t\left(X\right)\le \alpha$ and $|X|\le {2}^{\alpha }$, then $\exists p\in U\left(\alpha \right)$ ($X$ is a FU($p$)-space).

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