# Article

 Title: Weak-bases and $D$-spaces (English) Author: Burke, Dennis K. Language: English Journal: Commentationes Mathematicae Universitatis Carolinae ISSN: 0010-2628 (print) ISSN: 1213-7243 (online) Volume: 48 Issue: 2 Year: 2007 Pages: 281-289 . Category: math . 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.} (English) Keyword: quotient map Keyword: symmetrizable space Keyword: weak-base Keyword: $w$-structure Keyword: $D$-space MSC: 54B15 MSC: 54D70 MSC: 54E25 MSC: 54E40 idZBL: Zbl 1199.54065 idMR: MR2338096 . Date available: 2009-05-05T17:02:56Z Last updated: 2012-05-01 Stable URL: http://hdl.handle.net/10338.dmlcz/119658 . Reference: [A1] Arhangel'skii A.V.: Mappings and spaces.Russian Math. Surveys 21 (1966), 115-162. MR 0227950 Reference: [A2] Arhangel'skii A.V.: $D$-spaces and finite unions.Proc. Amer. Math. Soc. 132 (2004), 2163-2170. Zbl 1045.54009, MR 2053991 Reference: [A3] Arhangel'skii A.V.: $D$-spaces and covering properties.Topology Appl. 146/147 (2005), 437-449. Zbl 1063.54013, MR 2107163 Reference: [ABuz] Arhangel'skii A.V., Buzyakova R.: Addition theorems and $D$-spaces.Comment. Math. Univ. Carolin. 43 (2002), 653-663. Zbl 1090.54017, MR 2045787 Reference: [BW] Borges C.R., Wehrly A.C.: A study of $D$-spaces.Topology Proc. 16 (1991), 7-15. Zbl 0787.54023, MR 1206448 Reference: [Buz1] Buzyakova R.: On $D$-property of strong $\Sigma$-spaces.Comment. Math. Univ. Carolin. 43.3 (2002), 493-495. Zbl 1090.54018, MR 1920524 Reference: [Buz2] Buzyakova R.: Hereditary $D$-property of function spaces over compacta.Proc. Amer. Math. Soc. 132 (2004), 3433-3439. Zbl 1064.54029, MR 2073321 Reference: [vDP] van Douwen E.K., Pfeffer W.: Some properties of the Sorgenfrey line and related spaces.Pacific J. Math. 81 (1979), 371-377. Zbl 0409.54011, MR 0547605 Reference: [D] Džamonja M.: On $D$-spaces and discrete families of sets.in: Set theory, DIMACS Ser. Discrete Math. Theoret. Comput. Sci., 58, Amer. Math. Soc., Providence, RI, 2002, pp.45-63. MR 1903849 Reference: [FS] Fleissner W., Stanley A.: $D$-spaces.Topology Appl. 114 (2001), 261-271. Zbl 0983.54024, MR 1838325 Reference: [G] Gruenhage G.: A note on $D$-spaces.Topology Appl. 153 (2005-2006), 2218-2228. Zbl 1101.54029, MR 2238726 .

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