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| In [[mathematics]], a '''Cantor space''', named for [[Georg Cantor]], is a [[topology|topological]] 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<sup>ω</sup> is called "the" Cantor space. Note that, commonly, 2<sup>ω</sup> is referred to simply as the Cantor set, while the term Cantor space is reserved for the more general construction of ''D''<sup>S</sup> for a finite set ''D'' and a set ''S'' which might be finite, countable or possibly uncountable.<ref>Stephen Willard, ''General Topology'' (1970) Addison-Wesley Publishing. ''See section 17.9a''</ref>
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| == Examples ==
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| The Cantor set itself is a Cantor space. But the canonical example of a Cantor space is the [[countably infinite]] [[product topology|topological product]] of the [[discrete 2-point space]] {0, 1}. This is usually written as <math>2^\mathbb{N}</math> or 2<sup>ω</sup> (where 2 denotes the 2-element set {0,1} with the discrete topology). A point in 2<sup>ω</sup> is an infinite binary sequence, that is a sequence which assumes only the values 0 or 1. Given such a sequence ''a''<sub>0</sub>, ''a''<sub>1</sub>, ''a''<sub>2</sub>,..., one can map it to the real number
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| :<math>
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| \sum_{n=0}^\infty \frac{2 a_n}{3^{n+1}}.
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| </math>
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| This mapping gives a homeomorphism from 2<sup>ω</sup> onto the [[Cantor set]], demonstrating that 2<sup>ω</sup> is indeed a Cantor space.
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| Cantor spaces occur abundantly in [[real analysis]]. For example, they exist as subspaces in every perfect, [[complete space|complete]] [[metric space]]. (To see this, note that in such a space, any non-empty perfect set contains two disjoint non-empty perfect subsets of arbitrarily small diameter, and so one can imitate the construction
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| of the usual [[Cantor set]].) Also, every uncountable,
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| [[separable space|separable]], completely metrizable space contains
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| Cantor spaces as subspaces. This includes most of the common type of spaces in real analysis.
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| == Characterization ==
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| A topological characterization of Cantor spaces is given by [[Luitzen Egbertus Jan Brouwer|Brouwer]]'s theorem:{{citation needed|date=February 2014}}
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| :''Any two non-empty [[compact set|compact]] [[Hausdorff space]]s without [[isolated point]]s and having countable [[base (topology)|base]]s consisting of [[clopen set]]s are homeomorphic to each other''.
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| The topological property of having a base consisting of clopen sets is sometimes known as "zero-dimensionality". Brouwer's theorem can be restated as:
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| :''A topological space is a Cantor space if and only if it is non-empty, [[perfect space|perfect]], [[compact space|compact]], [[totally disconnected]], and [[metrizable]].''
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| This theorem is also equivalent (via [[Stone's representation theorem for Boolean algebras]]) to the fact that any two countable atomless [[Boolean algebra (structure)|Boolean algebra]]s are isomorphic.
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| == Properties ==
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| 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<sup>ω</sup>, because its construction as a product makes it amenable to analysis.
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| Cantor spaces have the following properties:
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| * The [[cardinality]] of any Cantor space is <math>2^{\aleph_0}</math>, that is, the [[cardinality of the continuum]].
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| * 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 curve]]s.
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| * A Hausdorff topological space is compact metrizable if and only if it is a continuous image of a Cantor space.<ref>N.L. Carothers, ''A Short Course on Banach Space Theory'', London Mathematical Society Student Texts '''64''', (2005) Cambridge University Press. ''See Chapter 12''</ref><ref>Willard, ''op.cit.'', ''See section 30.7''</ref>
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| Let ''C(X)'' denote the space of all real-valued, bounded 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:
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| * C(K) is [[isometry|isometric]] to a closed subspace of C(Δ).<ref>Carothers, ''op.cit.''</ref>
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| In general, this isometry is not unique, and thus is not properly a [[universal property]] in the categorical sense.
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| *The group of all [[homeomorphisms]] of the Cantor space is [[simple group|simple]].<ref>R.D. Anderson, ''The Algebraic Simplicity of Certain Groups of Homeomorphisms'', American Journal of Mathematics '''80''' (1958), pp. 955-963.</ref>
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| ==See also==
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| *[[Cantor cube]]
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| *[[Georg Cantor]]
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| ==References==
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| <references/>
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| *{{cite book | author=Kechris, A. | title= Classical Descriptive Set Theory | publisher=Springer | year=1995 | isbn = 0-387-94374-9| edition=Graduate Texts in Mathematics 156}}
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| {{DEFAULTSORT:Cantor Space}}
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| [[Category:Topological spaces]]
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| [[Category:Descriptive set theory]]
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