Second-Countable Space is First-Countable/Proof 1

Theorem

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

Then $T$ is also first-countable.

Proof

By definition $T$ is second-countable if and only if its topology has a countable basis.

Consider the entire set $S$ as an open set.

From Set is Open iff Neighborhood of all its Points, $S$ has that property.

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.

$\blacksquare$