Cantor set

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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.