Remark 2

There are spaces in which every point has a countable fundamental system of neighbourhoods but in which there exist closed sets which are not countable intersections of open sets (Exercise 15); such spaces are not metrizable.

Corollary 2 of Theorem 1, no. 4, shows that a countable product of metrizable topological spaces is metrizable. Also the sum $X$ (Chapter I, § 2, no. 4) of an arbitrary family $(X_i)_{i \in I}$ of metrizable spaces is metrizable. For if $d_i$ is a metric compatible with the topology of $X_i$ for each $i \in I$, we may assume that $d_i$ is bounded and that the diameter of $X_i$ is $\leq 1$; we can then define a distance $d$ compatible with the topology of $X$ by putting $d(x, y) = d_i(x, y)$ if $x$ and $y$ both belong to the same $X_i$, and $d(x, y) = 1$ otherwise.