Countable Complement Topology is Expansion of Finite Complement Topology

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

Let $T' = \left({S, \tau'}\right)$ be the countable complement topology on the same infinite set $S$.

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

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

Then $\complement_S \left({U}\right)$ is finite by definition of countable 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.