\begin{definition}[Definition:Cantor Set/Ternary Representation]
Consider the closed interval $\closedint 0 1 \subset \R$.
The '''Cantor set''' $\CC$ consists of all the points in $\closedint 0 1$ which can be expressed in base $3$ without using the digit $1$.
From Representation of Ternary Expansions, if any number has two different ternary representations, for example:
:$\dfrac 1 3 = 0.10000 \ldots = 0.02222$
then at most one of these can be written without any $1$'s in it.
Therefore this representation of points of $\CC$ is unique.
\end{definition}