Abstract: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. \vskip \smallskipamount \par \noindent {Theorem.} {Any symmetrizable space $X$ is a $D$-space $($hereditarily$)$.} \vskip \smallskipamount \par 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 are $D$-spaces so open $s$-images of metric spaces are already known to be $D$-spaces. \par A collection $\Cal W$ of subsets of a sequential space $X$ is said to be a {$w$-system} for the topology if whenever $x\in U\subseteq X$, with $U$ open, there exists a subcollection $\Cal V\subseteq \Cal W$ such that $x\in \bigcap \Cal V$, $\bigcup \Cal V$ is a weak-neighborhood of $x$, and $\bigcup \Cal V\subseteq U$. \vskip \smallskipamount \par \noindent {Theorem.} {A sequential space $X$ with a point-countable $w$-system is a $D$-space.} \vskip \smallskipamount \par \noindent {Corollary.} {A space $X$ with a point-countable weak-base is a $D$-space.} \vskip \smallskipamount \par \noindent {Corollary.} {Any $T_2$ quotient $s$-image of a metric space is a $D$-space.}
Keywords: quotient map, symmetrizable space, weak-base, $w$-structure, $D$-space
AMS Subject Classification: Primary 54B15; Secondary 54D70, 54E25, 54E40