Cardinality of the family of dense subsets of the reals
= Cardinality of the family of dense subsets of the reals
{title2=$2^{2^{\aleph_0}}$}
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.