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.
展开▼
机译：我们研究的拓扑空间可以表示为密集的可度量子空间的有限集合的并集。子空间在联合中密集的假设在下面起着至关重要的作用。尤其是，Example〜3.1显示了一个超紧缩空间$ X $，它是两个密集的可度量子空间的并集，而不必是$ p $空间。但是，如果正常空间$ X $是有限族$ mu $的密集子空间的并集，则每个密子空间都可以用一个完整的度量度量，那么$ X $也可以用一个完整的度量（定理〜2.6）进行度量。通过在最后一节中证明，如果空间的Lindel “ $ X $是有限家庭$ mu $的可密可量化子空间的并集，则$ X $是可分离且可量化的，我们还回答了MV〜Matveev的问题。 。
展开▼
