# Cantor set

The **Cantor set** is a set that may be generated by removing the middle third of a line segment on each iteration. It is a fractal with a Hausdorff dimension of ln(2)/ln(3), which is approximately 0.63.

## Topological properties

The Cantor set may be considered a topological space, 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

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

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

where 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

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.