Cantor space

 ( Topological Spaces ) 

In mathematics, a Cantor space, named for Georg Cantor, is a topology abstraction of the classical Cantor set: a topological space is a Cantor space if it is homeomorphic to the Cantor set. In set theory, the topological space 2^ω is called "the" Cantor space.

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Examples The Cantor set itself is a Cantor space. But the canonical example of a Cantor space is the countably infinite product topology of the discrete 2-point space {0, 1}. This is usually written as $2^\backslash mathbb\{N\}$ or 2^ω (where 2 denotes the 2-element set {0,1} with the discrete topology). A point in 2^ω is an infinite binary sequence, that is a sequence that assumes only the values 0 or 1. Given such a sequence a₀, a₁, a₂,..., one can map it to the real number

$\backslash sum\_\{n=0\}^\backslash infty\; \backslash frac\{2\; a\_n\}\{3^\{n+1\}\}.$

This mapping gives a homeomorphism from 2^ω onto the Cantor set, demonstrating that 2^ω is indeed a Cantor space.

Cantor spaces occur abundantly in real analysis. For example, they exist as subspaces in every perfect set, complete metric space. (To see this, note that in such a space, any empty set perfect set contains two disjoint sets non-empty perfect subsets of arbitrarily small diameter, and so one can imitate the construction of the usual Cantor set.) Also, every uncountable, separable space, completely metrizable space contains Cantor spaces as subspaces. This includes most of the common spaces in real analysis.

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Characterization A topological characterization of Cantor spaces is given by Brouwer's theorem:.

The topological property of having a base consisting of clopen sets is sometimes known as "zero-dimensional". Brouwer's theorem can be restated as:

This theorem is also equivalent (via Stone's representation theorem for Boolean algebras) to the fact that any two countable atomless Boolean algebras are isomorphic.

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Properties As can be expected from Brouwer's theorem, Cantor spaces appear in several forms. But many properties of Cantor spaces can be established using 2^ω, because its construction as a product makes it amenable to analysis.

Cantor spaces have the following properties:

• The cardinality of any Cantor space is $2^\{\backslash aleph\_0\}$, that is, the cardinality of the continuum.
• The product of two (or even any finite or countable number of) Cantor spaces is a Cantor space. Along with the Cantor function, this fact can be used to construct space-filling curves.
• A (non-empty) Hausdorff topological space is compact metrizable if and only if it is a continuous image of a Cantor space.N.L. Carothers, A Short Course on Banach Space Theory, London Mathematical Society Student Texts 64, (2005) Cambridge University Press. See Chapter 12Willard, op.cit., See section 30.7

Let C( X) denote the space of all real-valued, bounded function continuous functions on a topological space X. Let K denote a compact metric space, and Δ denote the Cantor set. Then the Cantor set has the following property:

• C( K) is isometry to a closed set subspace of C(Δ).Carothers, op.cit.

In general, this isometry is not unique, and thus is not properly a universal property in the category theory sense.

• The group of all homeomorphisms of the Cantor space is simple group.R.D. Anderson, The Algebraic Simplicity of Certain Groups of Homeomorphisms, American Journal of Mathematics 80 (1958), pp. 955-963.

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See also

• Space (mathematics)
• Cantor set
• Cantor cube

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External links

• Kechris, A. (1995). 9780387943749, Springer. . ISBN 9780387943749

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