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 metric. 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.

The Cantor set is uncountable, compact, second countable, dense-in-itself, 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 complete 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.