Countable Complement Topology is Expansion of Finite Complement Topology

Theorem
Let $T = \struct {S, \tau}$ be the countable complement topology on an infinite set $S$.

Let $T' = \struct {S, \tau'}$ be the finite complement topology on the same infinite set $S$.

Then $\tau$ is an expansion of $\tau'$.

Proof
Let $U \in \tau', U \ne \O$.

Then $\relcomp S U$ is finite by definition of finite complement topology.

Then by definition of countable complement topology, we have that $U \in \tau$.

So $\tau' \subseteq \tau$ and hence the result by definition of expansion.