Definition:Cantor Set/Ternary Representation
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Definition
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.
Sources
- 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.) ... (previous) ... (next): Part $\text {II}$: Counterexamples: $29$. The Cantor Set
- 2005: René L. Schilling: Measures, Integrals and Martingales ... (previous) ... (next): $\S 7$: Problem $10 \ \text{(vi)}$