Cantor set

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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.

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