Addition theorems for dense subspaces

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.