Maximum Cardinality of Separable Hausdorff Space

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Let $T = \struct {S, \tau}$ be a Hausdorff space which is separable.

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


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:

$\card S \le \card {2^{\powerset D} } = 2^{2^{\aleph_0} }$