Definition:Cantor Set

Definition
Define:


 * $\displaystyle A_n := \bigcup_{i=1}^{\dfrac{3^n - 1} 2} \left({\dfrac{2i-1}{3^n} . . \dfrac{2i}{3^n}}\right)$

Since $3^n$ is always odd, $\dfrac{3^n-1} 2$ is always an integer, and hence this union will always be perfectly defined.

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

Define:
 * $\mathfrak C_n := \left[{0 . .1}\right] - A_n$

The Cantor Set $\mathfrak C$ is defined as:
 * $\displaystyle \mathfrak C = \bigcap_{i=1}^\infty \mathfrak C_i$

Comments
The Cantor set is an well-known example in analysis.

It has several properties that make it interesting: it is closed, compact, uncountable, measure zero, perfect, nowhere dense, totally disconnected and fractal.

For more information, see Properties of the Cantor Set.