Cardinality of the family of dense subsets of the reals ($2^{2^{{\aleph }_0}}$)

There are $2^{\mathfrak{c}}$ dense subsets of $\mathbb{R}$, where $\mathfrak{c}=|\mathbb{R}|$. For a fixed Cantor set $C$ of size $\mathfrak{c}$, its complement $D$ is dense, and $A\mapsto D\cup A$ injects $\mathcal{P}(C)$ into the family of dense subsets. The reverse injection is the inclusion into $\mathcal{P}(\mathbb{R})$. The Cantor-Schröder-Bernstein theorem gives equality.