Second-Countable Space is Lindelöf

Theorem
Let $T = \left({S, \tau}\right)$ 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 $\mathcal B$ be this countable basis.

Let $\mathcal C$ be an open cover of $T$.

Every set in $\mathcal C$ is the union of a subset of $\mathcal B$.

So $\mathcal C$ itself is the union of a subset of $\mathcal B$.

This union of a subset of $\mathcal B$ is therefore a countable subcover of $\mathcal C$.

That is, $T$ is by definition Lindelöf.