A. V. Arhangel'skii
Addition theorems for dense subspaces

Comment.Math.Univ.Carolin. 56,4 (2015) 531-541.

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 Lindel\"of space $X$ is the union of a finite family $\mu $ of dense metrizable subspaces, then $X$ is separable and metrizable.

Keywords: dense subspace; perfect space; Moore space; \v Cech-complete; $p$-space; $\sigma $-disjoint base; uniform base; pseudocompact; point-countable base; pseudo-$\omega _1$-compact

DOI: DOI 10.14712/1213-7243.2015.142
AMS Subject Classification: 54A25 54B05