Definition:Cantor Set

Define

$$ A_n = \bigcup_{i=1}^{ \tfrac{3^n -1}{2 } } (\tfrac{2i-1}{3^n}, \tfrac{2i}{3^n}) $$

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

Consider the interval $$[0,1] \subset \mathbb{R} \ $$. Define

$$\mathfrak{C}_n = [0,1] - A_n \ $$

The Cantor Set $$\mathfrak{C} \ $$ is defined as:

$$\mathfrak{C} = \bigcap_{i=1}^\infty \mathfrak{C}_i \ $$.

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.