Existence of Topological Space satisfying Countable Chain Condition which is not Separable

Theorem
There exists at least one example of a topological space which satisfies the countable chain condition which is not also a separable space.

Proof
Let $T$ be a countable complement topology on an uncountable set.

From Countable Complement Space Satisfies Countable Chain Condition, $T$ satisfies the countable chain condition.

From Countable Complement Space is not Separable, $T$ is not a separable space.

Hence the result.