| Title:
|
Some relative properties on normality and paracompactness, and their absolute embeddings (English) |
| Author:
|
Kawaguchi, Shinji |
| Author:
|
Sokei, Ryoken |
| Language:
|
English |
| Journal:
|
Commentationes Mathematicae Universitatis Carolinae |
| ISSN:
|
0010-2628 (print) |
| ISSN:
|
1213-7243 (online) |
| Volume:
|
46 |
| Issue:
|
3 |
| Year:
|
2005 |
| Pages:
|
475-495 |
| . |
| Category:
|
math |
| . |
| Summary:
|
Paracompactness ($=2$-paracompactness) and normality of a subspace $Y$ in a space $X$ defined by Arhangel'skii and Genedi [4] are fundamental in the study of relative topological properties ([2], [3]). These notions have been investigated by primary using of the notion of weak $C$- or weak $P$-embeddings, which are extension properties of functions defined in [2] or [18]. In fact, Bella and Yaschenko [8] characterized Tychonoff spaces which are normal in every larger Tychonoff space, and this result is essentially implied by their previous result in [8] on a corresponding case of weak $C$-embeddings. In this paper, we introduce notions of $1$-normality and $1$-collectionwise normality of a subspace $Y$ in a space $X$, which are closely related to $1$-paracompactness of $Y$ in $X$. Furthermore, notions of quasi-$C^\ast$- and quasi-$P$-embeddings are newly defined. Concerning the result of Bella and Yaschenko above, by characterizing absolute cases of quasi-$C^*$- and quasi-$P$-embeddings, we obtain the following result: a Tychonoff space $Y$ is $1$-normal (or equivalently, $1$-collectionwise normal) in every larger Tychonoff space if and only if $Y$ is normal and almost compact. As another concern, we also prove that a Tychonoff (respectively, regular, Hausdorff) space $Y$ is $1$-metacompact in every larger Tychonoff (respectively, regular, Hausdorff) space if and only if $Y$ is compact. Finally, we construct a Tychonoff space $X$ and a subspace $Y$ such that $Y$ is $1$-paracompact in $X$ but not $1$-subparacompact in $X$. This is a negative answer to a question of Qu and Yasui in [25]. (English) |
| Keyword:
|
$1$-paracompactness of $Y$ in $X$ |
| Keyword:
|
$2$-paracompactness of $Y$ in $X$ |
| Keyword:
|
$1$-collectionwise normality of $Y$ in $X$ |
| Keyword:
|
$2$-collectionwise normality of $Y$ in $X$ |
| Keyword:
|
$1$-normality of $Y$ in $X$ |
| Keyword:
|
$2$-normality of $Y$ in $X$ |
| Keyword:
|
quasi-$P$-embedding |
| Keyword:
|
quasi-$C$-embedding |
| Keyword:
|
quasi-$C^{*}$-embedding |
| Keyword:
|
$1$-metacompactness of $Y$ in $X$ |
| Keyword:
|
$1$-subparacompactness of $Y$ in $X$ |
| MSC:
|
54B05 |
| MSC:
|
54B10 |
| MSC:
|
54C20 |
| MSC:
|
54C45 |
| MSC:
|
54D15 |
| MSC:
|
54D20 |
| idZBL:
|
Zbl 1121.54018 |
| idMR:
|
MR2174526 |
| . |
| Date available:
|
2009-05-05T16:52:39Z |
| Last updated:
|
2012-04-30 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/119542 |
| . |
| Reference:
|
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| Reference:
|
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| . |
