# Addition theorems for dense subspaces

• Volume: 56, Issue: 4, page 531-541
• ISSN: 0010-2628

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## Abstract

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We study topological spaces that can be represented as the union of a finite collection of dense metrizable subspaces. The assumption that the subspaces are dense in the union plays a crucial role below. In particular, Example 3.1 shows that a paracompact space $X$ which is the union of two dense metrizable subspaces need not be a $p$-space. However, if a normal space $X$ is the union of a finite family $\mu$ of dense subspaces each of which is metrizable by a complete metric, then $X$ is also metrizable by a complete metric (Theorem 2.6). We also answer a question of M.V. Matveev by proving in the last section that if a Lindelöf space $X$ is the union of a finite family $\mu$ of dense metrizable subspaces, then $X$ is separable and metrizable.

## How to cite

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Arhangel'skii, Aleksander V.. "Addition theorems for dense subspaces." Commentationes Mathematicae Universitatis Carolinae 56.4 (2015): 531-541. <http://eudml.org/doc/276210>.

@article{Arhangelskii2015,
abstract = {We study topological spaces that can be represented as the union of a finite collection of dense metrizable subspaces. The assumption that the subspaces are dense in the union plays a crucial role below. In particular, Example 3.1 shows that a paracompact space $X$ which is the union of two dense metrizable subspaces need not be a $p$-space. However, if a normal space $X$ is the union of a finite family $\mu$ of dense subspaces each of which is metrizable by a complete metric, then $X$ is also metrizable by a complete metric (Theorem 2.6). We also answer a question of M.V. Matveev by proving in the last section that if a Lindelöf space $X$ is the union of a finite family $\mu$ of dense metrizable subspaces, then $X$ is separable and metrizable.},
author = {Arhangel'skii, Aleksander V.},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {dense subspace; perfect space; Moore space; Čech-complete; $p$-space; $\sigma$-disjoint base; uniform base; pseudocompact; point-countable base; pseudo-$\omega _1$-compact},
language = {eng},
number = {4},
pages = {531-541},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {Addition theorems for dense subspaces},
url = {http://eudml.org/doc/276210},
volume = {56},
year = {2015},
}

TY - JOUR
AU - Arhangel'skii, Aleksander V.
TI - Addition theorems for dense subspaces
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2015
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 56
IS - 4
SP - 531
EP - 541
AB - We study topological spaces that can be represented as the union of a finite collection of dense metrizable subspaces. The assumption that the subspaces are dense in the union plays a crucial role below. In particular, Example 3.1 shows that a paracompact space $X$ which is the union of two dense metrizable subspaces need not be a $p$-space. However, if a normal space $X$ is the union of a finite family $\mu$ of dense subspaces each of which is metrizable by a complete metric, then $X$ is also metrizable by a complete metric (Theorem 2.6). We also answer a question of M.V. Matveev by proving in the last section that if a Lindelöf space $X$ is the union of a finite family $\mu$ of dense metrizable subspaces, then $X$ is separable and metrizable.
LA - eng
KW - dense subspace; perfect space; Moore space; Čech-complete; $p$-space; $\sigma$-disjoint base; uniform base; pseudocompact; point-countable base; pseudo-$\omega _1$-compact
UR - http://eudml.org/doc/276210
ER -

## References

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