# Article

Full entry | PDF   (0.2 MB)
Keywords:
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.}
References:
[A1] Arhangel'skii A.V.: Mappings and spaces. Russian Math. Surveys 21 (1966), 115-162. MR 0227950
[A2] Arhangel'skii A.V.: $D$-spaces and finite unions. Proc. Amer. Math. Soc. 132 (2004), 2163-2170. MR 2053991 | Zbl 1045.54009
[A3] Arhangel'skii A.V.: $D$-spaces and covering properties. Topology Appl. 146/147 (2005), 437-449. MR 2107163 | Zbl 1063.54013
[ABuz] Arhangel'skii A.V., Buzyakova R.: Addition theorems and $D$-spaces. Comment. Math. Univ. Carolin. 43 (2002), 653-663. MR 2045787 | Zbl 1090.54017
[BW] Borges C.R., Wehrly A.C.: A study of $D$-spaces. Topology Proc. 16 (1991), 7-15. MR 1206448 | Zbl 0787.54023
[Buz1] Buzyakova R.: On $D$-property of strong $\Sigma$-spaces. Comment. Math. Univ. Carolin. 43.3 (2002), 493-495. MR 1920524 | Zbl 1090.54018
[Buz2] Buzyakova R.: Hereditary $D$-property of function spaces over compacta. Proc. Amer. Math. Soc. 132 (2004), 3433-3439. MR 2073321 | Zbl 1064.54029
[vDP] van Douwen E.K., Pfeffer W.: Some properties of the Sorgenfrey line and related spaces. Pacific J. Math. 81 (1979), 371-377. MR 0547605 | Zbl 0409.54011
[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
[FS] Fleissner W., Stanley A.: $D$-spaces. Topology Appl. 114 (2001), 261-271. MR 1838325 | Zbl 0983.54024
[G] Gruenhage G.: A note on $D$-spaces. Topology Appl. 153 (2005-2006), 2218-2228. MR 2238726 | Zbl 1101.54029

Partner of