Second-Countability is Hereditary

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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.

$\blacksquare$


Sources