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Keywords:
quotient map; symmetrizable space; weak-base; $w$-structure; $D$-space
quotient map; symmetrizable space; weak-base; $w$-structure; $D$-space
Summary:
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. \smallskip \noindent {\bf Theorem.} {\sl Any symmetrizable space $X$ is a $D$-space $($hereditarily$)$.} \smallskip 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. A collection $\Cal W$ of subsets of a sequential space $X$ is said to be a {\it $w$-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 $\bigcup \Cal V\subseteq U$. \smallskip \noindent {\bf Theorem.} {\sl A sequential space $X$ with a point-countable $w$-system is a $D$-space.} \smallskip \noindent {\bf Corollary.} {\sl A space $X$ with a point-countable weak-base is a $D$-space.} \smallskip \noindent {\bf Corollary.} {\sl Any $T_2$ quotient $s$-image of a metric space is a $D$-space.}
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