Compact Complement Topology is Connected

Theorem

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

Then $T$ is a connected space.

Proof

Follows from:

Compact Complement Topology is Irreducible

Irreducible Space is Connected

$\blacksquare$

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: $3$