Definition:Cantor Set/Ternary Representation
Consider the closed interval $\closedint 0 1 \subset \R$.
The Cantor set $\mathcal C$ consists of all the points in $\closedint 0 1$ which can be expressed in base $3$ without using the digit $1$.
- $\dfrac 1 3 = 0.10000 \ldots = 0.02222$
then at most one of these can be written without any $1$'s in it.
Therefore this representation of points of $\mathcal C$ is unique.