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 $p\le {}_{RK}q$ ($\le {}_{RK}$ denotes the Rudin–Keisler order) $\iff$ every $p$-sequential space is $q$-sequential $\iff$ every FU($p$)-space is a FU($q$)-space. We generalize the spaces $S_n$ to construct examples of...

It is shown that a space $X$ is $L\left({}^{\mu }p\right)$-Weakly Fréchet-Urysohn for $p\in {\omega }^{*}$ iff it is $L\left({}^{\nu }p\right)$-Weakly Fréchet-Urysohn for arbitrary $\mu ,\nu <{\omega }_1$, where ${}^{\mu }p$ is the $\mu$-th left power of $p$ and $L\left(q\right)=\{{}^{\mu }q:\mu <{\omega }_1\}$ for $q\in {\omega }^{*}$. We also prove that for $p$-compact spaces, $p$-sequentiality and the property of being a $L\left({}^{\nu }p\right)$-Weakly Fréchet-Urysohn space with $\nu <{\omega }_1$, are equivalent; consequently if $X$ is $p$-compact and $\nu <{\omega }_1$, then $X$ is $p$-sequential iff $X$ is ${}^{\nu }p$-sequential (Boldjiev and Malyhin gave, for each $P$-point $p\in {\omega }^{*}$, an example of a compact space $X_p$ which is ${}^{2}p$-Fréchet-Urysohn...