First-Countability is Hereditary

Theorem
Let $T = \left({X, \vartheta}\right)$ be a topological space which is first-countable.

Let $T_H = \left({H, \vartheta_H}\right)$, where $\varnothing \subset H \subseteq X$, be a subspace of $T$.

Then $T_H$ is first-countable.

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

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

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