Compact Complement Topology is First-Countable

Theorem

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

Then $T$ is a first-countable space.

Proof

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.

$\blacksquare$

Sources

• 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.) ... (previous) ... (next): Part $\text {II}$: Counterexamples: $22$. Compact Complement Topology: $5$