| Title:
|
A note on transitively $D$-spaces (English) |
| Author:
|
Peng, Liang-Xue |
| Language:
|
English |
| Journal:
|
Czechoslovak Mathematical Journal |
| ISSN:
|
0011-4642 (print) |
| ISSN:
|
1572-9141 (online) |
| Volume:
|
61 |
| Issue:
|
4 |
| Year:
|
2011 |
| Pages:
|
1049-1061 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
In this note, we show that if for any transitive neighborhood assignment $\phi $ for $X$ there is a point-countable refinement ${\mathcal F}$ such that for any non-closed subset $A$ of $X$ there is some $V\in {\mathcal F}$ such that $|V\cap A|\geq \omega $, then $X$ is transitively $D$. As a corollary, if $X$ is a sequential space and has a point-countable $wcs^*$-network then $X$ is transitively $D$, and hence if $X$ is a Hausdorff $k$-space and has a point-countable $k$-network, then $X$ is transitively $D$. We prove that if $X$ is a countably compact sequential space and has a point-countable $wcs^*$-network, then $X$ is compact. We point out that every discretely Lindelöf space is transitively $D$. Let $(X, \tau )$ be a space and let $(X, {\mathcal T})$ be a butterfly space over $(X, \tau )$. If $(X, \tau )$ is Fréchet and has a point-countable $wcs^*$-network (or is a hereditarily meta-Lindelöf space), then $(X, {\mathcal T})$ is a transitively $D$-space. (English) |
| Keyword:
|
transitively $D$ |
| Keyword:
|
sequential |
| Keyword:
|
discretely Lindelöf |
| Keyword:
|
$wcs^*$-network |
| MSC:
|
54D20 |
| MSC:
|
54F99 |
| MSC:
|
54G99 |
| idZBL:
|
Zbl 1249.54054 |
| idMR:
|
MR2886256 |
| DOI:
|
10.1007/s10587-011-0047-5 |
| . |
| Date available:
|
2011-12-16T15:47:26Z |
| Last updated:
|
2020-07-03 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/141806 |
| . |
| Reference:
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