Abstract: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 $f(\alpha )\subseteq A$ 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 $p \leq { _{RK}} q$ ($\leq { _{RK}}$ 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 $\Leftrightarrow \forall \nu <\omega _1 (p^\nu \leq { _{RK}} q)$, for $p,q \in \omega ^*$; and $S_n$ is a {FU($p$)}-space for $p\in \omega ^*$ and $1<n<\omega \Leftrightarrow $ every sequential space $X$ with $\sigma (X)\leq n$ is a {FU($p$)}-space $\Leftrightarrow \exists \{p_{n-2},..., p_1\}\subseteq \omega ^* (p_{n-2}<{ _{RK}}...<{ _{RK}} p_1 <_{ l} p)$; 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 (\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 {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 {FU($p$)}-space).
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
AMS Subject Classification: Primary 04A20, 54A25, 54D55; Secondary 54D99