Maximum Cardinality of Separable Hausdorff Space

Theorem
Let $T = \left({X, \vartheta}\right)$ be a Hausdorff space which is separable.

Then $X$ can have a cardinality no greater than $2^{2^{\aleph_0}}$.

Proof
Let $D$ be an everywhere dense subset of $X$ which is countable, as is guaranteed as $T$ is separable.