| 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 |
| . |
