Some relative properties on normality and paracompactness, and their absolute embeddings

Shinji Kawaguchi; Ryoken Sokei

Commentationes Mathematicae Universitatis Carolinae (2005)

  • Volume: 46, Issue: 3, page 475-495
  • ISSN: 0010-2628

Abstract

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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 * - 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].

How to cite

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Kawaguchi, Shinji, and Sokei, Ryoken. "Some relative properties on normality and paracompactness, and their absolute embeddings." Commentationes Mathematicae Universitatis Carolinae 46.3 (2005): 475-495. <http://eudml.org/doc/249572>.

@article{Kawaguchi2005,
abstract = {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].},
author = {Kawaguchi, Shinji, Sokei, Ryoken},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {$1$-paracompactness of $Y$ in $X$; $2$-paracompactness of $Y$ in $X$; $1$-collectionwise normality of $Y$ in $X$; $2$-collectionwise normality of $Y$ in $X$; $1$-normality of $Y$ in $X$; $2$-normality of $Y$ in $X$; quasi-$P$-embedding; quasi-$C$-embedding; quasi-$C^\{*\}$-embedding; $1$-metacompactness of $Y$ in $X$; $1$-subparacompactness of $Y$ in $X$; 1-paracompactness of in ; 2-paracompactness of in },
language = {eng},
number = {3},
pages = {475-495},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {Some relative properties on normality and paracompactness, and their absolute embeddings},
url = {http://eudml.org/doc/249572},
volume = {46},
year = {2005},
}

TY - JOUR
AU - Kawaguchi, Shinji
AU - Sokei, Ryoken
TI - Some relative properties on normality and paracompactness, and their absolute embeddings
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2005
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 46
IS - 3
SP - 475
EP - 495
AB - 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].
LA - eng
KW - $1$-paracompactness of $Y$ in $X$; $2$-paracompactness of $Y$ in $X$; $1$-collectionwise normality of $Y$ in $X$; $2$-collectionwise normality of $Y$ in $X$; $1$-normality of $Y$ in $X$; $2$-normality of $Y$ in $X$; quasi-$P$-embedding; quasi-$C$-embedding; quasi-$C^{*}$-embedding; $1$-metacompactness of $Y$ in $X$; $1$-subparacompactness of $Y$ in $X$; 1-paracompactness of in ; 2-paracompactness of in
UR - http://eudml.org/doc/249572
ER -

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