Cantor set

The Cantor set is a topological space which may be obtained as a fractal generated by removing the middle third of a line segment on each iteration: as such it has a Hausdorff dimension of ln(2)/ln(3), which is approximately 0.63.

Topological properties

The Cantor set is homeomorphic to a product of countably many copies of a two-point space with the discrete topology. It is thus compact. It may be realised as the space of binary sequences

$C=\left\lbrace (x_{n})_{n\in \mathbf {N} }:x_{n}\in \{0,1\}\right\rbrace ,\,$

in which the open sets are generated by the cylinders, of the form

$C_{s}=\left\lbrace (x_{n})\in C:x_{n}=s_{n}{\mbox{ for }}n=0,\ldots ,k-1\right\rbrace ,\,$

where s is a given binary sequence of length k.

As a topological space, the Cantor set is uncountable, compact, second countable and totally disconnected.

Metric properties

The topology on the countable product of the two-point space D is induced by the metric

$d(\mathbf {x} ,\mathbf {y} )=\sum _{m=0}^{\infty }d_{2}(x_{n},y_{n}).2^{-n}\,$

where $d_{2}$ is the discrete metric on D.

The Cantor set is a complete metric space with respect to d.

Embedding in the unit interval

The Cantor set may be embedded in the unit interval by the map

$f:\mathbf {x} \mapsto \sum _{n=0}^{\infty }2x_{n}.3^{-n-1}$

which is a homeomorphism onto the subset of the unit interval obtained by iteratively deleting the middle third of each interval. As a subset of the unit interval it is closed, nowhere dense, perfect and dense-in-itself. It has Lebesgue measure zero.