Weak-bases and $D$-spaces

Burke, Dennis K.. "Weak-bases and $D$-spaces." Commentationes Mathematicae Universitatis Carolinae 48.2 (2007): 281-289. <http://eudml.org/doc/250236>.

@article{Burke2007,
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. Theorem.Any symmetrizable space $X$ is a $D$-space $($hereditarily$)$. 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 $\mathcal \{W\}$ 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 $\mathcal \{V\}\subseteq \mathcal \{W\}$ such that $x\in \bigcap \mathcal \{V\}$, $\bigcup \mathcal \{V\}$ is a weak-neighborhood of $x$, and $\bigcup \mathcal \{V\}\subseteq U$. Theorem.A sequential space $X$ with a point-countable $w$-system is a $D$-space.Corollary.A space $X$ with a point-countable weak-base is a $D$-space.Corollary.Any $T_2$ quotient $s$-image of a metric space is a $D$-space.},
author = {Burke, Dennis K.},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {quotient map; symmetrizable space; weak-base; $w$-structure; $D$-space; quotient map; symmetrizable space; weak-base; -structure; -space},
language = {eng},
number = {2},
pages = {281-289},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {Weak-bases and $D$-spaces},
url = {http://eudml.org/doc/250236},
volume = {48},
year = {2007},
}

TY - JOUR
AU - Burke, Dennis K.
TI - Weak-bases and $D$-spaces
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2007
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 48
IS - 2
SP - 281
EP - 289
AB - 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. Theorem.Any symmetrizable space $X$ is a $D$-space $($hereditarily$)$. 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 $\mathcal {W}$ 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 $\mathcal {V}\subseteq \mathcal {W}$ such that $x\in \bigcap \mathcal {V}$, $\bigcup \mathcal {V}$ is a weak-neighborhood of $x$, and $\bigcup \mathcal {V}\subseteq U$. Theorem.A sequential space $X$ with a point-countable $w$-system is a $D$-space.Corollary.A space $X$ with a point-countable weak-base is a $D$-space.Corollary.Any $T_2$ quotient $s$-image of a metric space is a $D$-space.
LA - eng
KW - quotient map; symmetrizable space; weak-base; $w$-structure; $D$-space; quotient map; symmetrizable space; weak-base; -structure; -space
UR - http://eudml.org/doc/250236
ER -