Definition:Cantor Set

Definition

As a Limit of Intersections

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

$k \left({n}\right) := \dfrac {3^n - 1} 2$

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

Since $3^n$ is always odd, $k \left({n}\right)$ is always an integer, and hence the union will always be perfectly defined.

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

Define:

$\mathcal C_n := \left[{0 \,.\,.\, 1}\right] \setminus A_n$

The Cantor Set $\mathcal C$ is defined as:

$\displaystyle \mathcal C = \bigcap_{i=1}^\infty \ \mathcal C_i$

From Ternary Representation

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

The Cantor set $\mathcal C$ consists of all the points in $\left[{0 .. 1}\right]$ 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, e.g.:

$\dfrac 1 3 = 0.10000 \ldots = 0.02222$

then at most one of these can be written without any $1$'s in it.

Therefore this representation of points of $\mathcal C$ is unique.

Cantor Space

The Cantor set is usually considered as a topological subspace of the real number space under the Euclidean topology.

Also known as

Some sources refer to the Cantor set as Cantor's discontinuum.

Comments

The Cantor set is a 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.

Also see

• Results about Cantor set can be found here.

Source of Name

This entry was named for Georg Cantor.