Maximum Cardinality of Separable Hausdorff Space

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

Then $S$ 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.

Consider the mapping $\Phi: S \to 2^{\mathcal P \left({D}\right)}$ defined as:
 * $\forall x \in S: \Phi \left({x}\right) \left({A}\right) = 1 \iff A = D \cap U_x$ for some neighborhood $U_x$ of $x$

It is seen that if $T$ is a Hausdorff space, then $\Phi$ is an injection.

It follows that:
 * $\left \lvert{S}\right\rvert \le \left \lvert{2^{\mathcal P \left({D}\right)}}\right\rvert = 2^{2^{\aleph_0}}$