# Compact Complement Topology is Sequentially Compact

## Theorem

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

Then $T$ is a sequentially compact space.

## Proof

We have:

Compact Complement Topology is Compact
Compact Space is Countably Compact
Compact Complement Topology is First-Countable
First-Countable Space is Sequentially Compact iff Countably Compact

$\blacksquare$