First-Countability is Hereditary

Theorem
Let $T = \struct {S, \tau}$ be a topological space which is first-countable.

Let $T_H = \struct {H, \tau_H}$, where $\O \subset H \subseteq S$, be a subspace of $T$.

Then $T_H$ is first-countable.

Proof
From the definition of first-countable, every point in $S$ has a countable local basis in $T$.

The intersection of $H$ with the countable local basis of $S$ provides a countable local basis for $H$.

As every point in $H$ is also a point in $S$, the result follows from Basis for Topological Subspace‎.