Second-Countable Space is First-Countable

Theorem
Let $T = \left({X, \vartheta}\right)$ be a topological space which is second-countable.

Then $T$ is also both first-countable and separable.

Proof
$T$ is second-countable iff its topology has a countable basis.

If we consider the entire set $X$ as an open set, then we see that $X$ is a neighborhood of all points.

As $T$ has a countable basis, then (trivially) every point in $T$ has a countable local basis.

So a second-countable space is trivially first-countable.

Now note that the union of one point from each basis element forms a countable dense subset.

So $T$ is by definition separable.