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Title: Addition theorems for dense subspaces (English)
Author: Arhangel'skii, A. V.
Language: English
Journal: Commentationes Mathematicae Universitatis Carolinae
ISSN: 0010-2628 (print)
ISSN: 1213-7243 (online)
Volume: 56
Issue: 4
Year: 2015
Pages: 531-541
Summary lang: English
Category: math
Summary: 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. (English)
Keyword: dense subspace
Keyword: perfect space
Keyword: Moore space
Keyword: Čech-complete
Keyword: $p$-space
Keyword: $\sigma $-disjoint base
Keyword: uniform base
Keyword: pseudocompact
Keyword: point-countable base
Keyword: pseudo-$\omega _1$-compact
MSC: 54A25
MSC: 54B05
idZBL: Zbl 06537722
idMR: MR3434227
DOI: 10.14712/1213-7243.2015.142
Date available: 2015-12-17T11:52:47Z
Last updated: 2018-01-04
Stable URL:
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