Second-Countability is Hereditary

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

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

Then $T_H$ is second-countable.

Proof
From the definition of second-countable, $\left({X, \vartheta}\right)$ has a countable basis.

That is, $\exists \mathcal B \subseteq \vartheta$ such that:
 * for all $U \in \vartheta$, $U$ is a union of sets from $\mathcal B$
 * $\mathcal B$ is countable.

As $H \subseteq X$ it follows that a $H$ itself is a union of sets from $\mathcal B$.

The result follows from Basis For Topological Subspace‎.