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.

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 $\BB$.

$\blacksquare$

1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.) ... (previous) ... (next): Part $\text I$: Basic Definitions: Section $1$: General Introduction: Countability Properties
2011: John M. Lee: Introduction to Topological Manifolds (2nd ed.) ... (previous) ... (next): $\S 3$: New Spaces From Old: Subspaces