Cantor set

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The Cantor set $\mathcal{C}$ is a subset of the real numbers that exhibits a number of interesting and counter-intuitive properties. It is among the simplest examples of a fractal.

The Cantor set can be described recursively as follows: begin with the set [0,1] and then remove the ( open) middle third, dividing the interval into two intervals of length $\frac{1}{3}$. Then remove the middle third of the two remaining segments, and remove the middle third of the four remaining segments, and so on ad infinitum.

Equivalently, we may define $\mathcal{C}$ to be the set of real numbers between $0$ and $1$ with a base three expansion that contains only the digits $0$ and $1$.

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