Existence of First-Countable Space which is not Second-Countable

Theorem
There exists at least one example of a first-countable topological space which is not also a second-countable space.

Proof
Let $T$ be a right half-open interval space.

From Right Half-Open Interval Space is First-Countable, $T$ is a first-countable space.

From Right Half-Open Interval Space is not Second-Countable, $T$ is not a second-countable space.

Hence the result.