Maximum Cardinality of Separable Hausdorff Space

Theorem
Let $T = \struct {S, \tau}$ 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 $S$ which is countable, as is guaranteed as $T$ is separable.

Consider the mapping $\Phi: S \to 2^{\powerset D}$ defined as:
 * $\forall x \in S: \map {\map \Phi x} A = 1 \iff A = D \cap U_x$ 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:
 * $\card S \le \card {2^{\powerset D} } = 2^{2^{\aleph_0} }$