Definition:Cantor Set/Ternary Representation

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Consider the closed interval $\left[{0 \,.\,.\, 1}\right] \subset \R$.

The Cantor set $\mathcal C$ consists of all the points in $\left[{0 \,.\,.\, 1}\right]$ 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 $\mathcal C$ is unique.