Compact Complement Topology is First-Countable

Theorem
Let $T = \left({\R, \tau}\right)$ be the compact complement topology on $\R$.

Then $T$ is a first-countable space.

Proof
Let $\mathcal C$ be an open cover of $\R$.

Let $p \in \R$.

Consider the set:


 * $\mathcal B_p = \left\{{\left({-\infty .. -n}\right) \cup \left({p - \dfrac 1 n .. p + \dfrac 1 n}\right) \cup \left({n .. \infty}\right): n \in \N}\right\}$

From Countable Local Basis in Compact Complement Topology, $\mathcal B_p$ is a countable local basis for $T$.

Hence the result, by definition of first-countable space.