Second-Countability is Hereditary

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

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

Then $T_H$ is second-countable.

Proof
From the definition of second-countable, $\struct {S, \tau}$ has a countable basis.

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

As $H \subseteq S$ it follows that a $H$ itself is a union of sets from $\BB$.

The result follows from Basis for Topological Subspace‎.