Arhangel’skiǐ proved that if $X$ and $Y$ are completely regular spaces such that $C_p\left(X\right)$ and $C_p\left(Y\right)$ are linearly homeomorphic, then $X$ is pseudocompact if and only if $Y$ is pseudocompact. In addition he proved the same result for compactness, $\sigma$-compactness and realcompactness. In this paper we prove that if $\phi :C_p\left(X\right)\to C_p\left(X\right)$ is a continuous linear surjection, then $Y$ is pseudocompact provided $X$ is and if $\phi$ is a continuous linear injection, then $X$ is pseudocompact provided $Y$ is. We also give examples that both statements...

It is shown that certain weak-base structures on a topological space give a $D$-space. This solves the question by A.V. Arhangel’skii of when quotient images of metric spaces are $D$-spaces. A related result about symmetrizable spaces also answers a question of Arhangel’skii. Hence, quotient mappings, with compact fibers, from metric spaces have a $D$-space image. What about quotient $s$-mappings? Arhangel’skii and Buzyakova have shown that spaces with a point-countable base...

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...