# Weak-bases and $D$-spaces

Commentationes Mathematicae Universitatis Carolinae (2007)

- Volume: 48, Issue: 2, page 281-289
- ISSN: 0010-2628

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topBurke, 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 -

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