Initial Topology Generated by Countable Family of Functions Separating Points is Metrizable

Theorem
Let $X$ be a set.

For each $n \in \N$, let $\struct {Y_n, d_n}$ be a metric space.

Let $\family {f_n}_{n \in \N}$ be a indexed family of functions such that:


 * for each $x, y \in X$ with $x \ne y$ there exists $n \in \N$ such that $\map {f_n} x = \map {f_n} y$.

Let $\tau$ be the initial topology on $X$ generated by $\family {f_n}_{n \in \N}$.

Then $\tau$ is metrizable.