Corollary

A subset $B$ of $A$ is dense in $A$ if and only if $\overline{B} = \overline{A}$ in $X$ (i.e. if and only if $A \subset \overline{B}$).

It follows that if $A, B, C$ are three subsets of $X$ such that $A \supset B \supset C$, and if $B$ is dense in $A$, and $C$ is dense in $B$, then $C$ is dense in $A$ (transitivity of density), for we have $\overline{A} = \overline{B} = \overline{C}$ in $X$.