Finite Complement Space is Connected

Theorem

Let $T = \struct {S, \tau}$ be a finite complement topology on an infinite set $S$.

Then $T$ is a connected space.

Proof

We have that a Finite Complement Space is Irreducible.

The result follows from 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: $18 \text { - } 19$. Finite Complement Topology: $9$