Compact Complement Topology is First-Countable

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Let $T = \struct {\R, \tau}$ be the compact complement topology on $\R$.

Then $T$ is a first-countable space.


Let $\CC$ be an open cover of $\R$.

Let $p \in \R$.

Consider the set:

$\BB_p = \set {\openint {-\infty} {-n} \cup \openint {p - \dfrac 1 n} {p + \dfrac 1 n} \cup \openint n \infty: n \in \N}$

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

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