Second-Countable Space is Lindelöf

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

Then $T$ is also a Lindelöf space.

Proof
Let $T$ be second-countable.

Then by definition its topology has a countable basis.

Let $\BB$ be this countable basis.

Let $\CC$ be an open cover of $T$.

Every set in $\CC$ is the union of a subset of $\BB$.

So $\CC$ itself is the union of a subset of $\BB$.

This union of a subset of $\BB$ is therefore a countable subcover of $\CC$.

That is, $T$ is by definition Lindelöf.