Category:Definitions/Cantor Set

This category contains definitions related to Cantor Set.
Related results can be found in Category:Cantor Set.

Define, for $n \in \N$, subsequently:

$\map k n := \dfrac {3^n - 1} 2$

$\ds A_n := \bigcup_{i \mathop = 1}^{\map k n} \openint {\frac {2 i - 1} {3^n} } {\frac {2 i} {3^n} }$

Since $3^n$ is always odd, $\map k n$ is always an integer, and hence the union will always be perfectly defined.

Consider the closed interval $\closedint 0 1 \subset \R$.

Define:

$\CC_n := \closedint 0 1 \setminus A_n$

The Cantor set $\CC$ is defined as:

$\ds \CC = \bigcap_{n \mathop = 1}^\infty \CC_n$