Editing Cantor space

{{Short description|Topological space}}
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.

== Examples ==

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,&thinsp;1}.  This is usually written as <math>2^\mathbb{N}</math> or 2<sup>ω</sup> (where 2 denotes the 2-element [[set (mathematics)|set]] {0,1} with the [[discrete topology]]). A point in 2<sup>ω</sup> is an infinite binary sequence, that is a sequence that 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]]
:<math>\sum_{n=0}^\infty \frac{2 a_n}{3^{n+1}}.</math>
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.

Cantor spaces occur abundantly in [[real analysis]]. For example, they exist as [[Subspace topology|subspace]]s in every [[perfect set|perfect]], [[complete metric space]].  (To see this, note that in such a space, any [[empty set|non-empty]] perfect set contains two [[disjoint sets|disjoint]] 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|separable]], [[completely metrizable space]] contains
Cantor spaces as subspaces.  This includes most of the common spaces in real analysis.

== Characterization ==

A topological characterization of Cantor spaces is given by [[Luitzen Egbertus Jan Brouwer|Brouwer]]'s theorem:<ref>{{citation|first=L. E. J.|last=Brouwer|authorlink=L. E. J. Brouwer|title=On the structure of perfect sets of points|journal=Proc. Koninklijke Akademie van Wetenschappen|volume=12|year=1910|pages=785–794|url=http://www.dwc.knaw.nl/DL/publications/PU00013496.pdf}}.</ref>
{{Math theorem
|name=Brouwer's theorem
|Any two non-empty [[compact space|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.}}
The topological property of having a base consisting of clopen sets is sometimes known as "[[zero-dimensional|zero-dimensionality]]". Brouwer's theorem can be restated as:
{{Math theorem|A topological space is a Cantor space [[if and only if]] it is non-empty, [[perfect set|perfect]], compact, [[totally disconnected]], and [[metrizable]].}}
This theorem is also equivalent (via [[Stone's representation theorem for Boolean algebras]]) to the fact that any two [[countable atomless Boolean algebra]]s are [[isomorphic]].

== 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<sup>ω</sup>, because its construction as a product makes it amenable to analysis.

Cantor spaces have the following properties:
* The [[cardinality]] of any Cantor space is <math>2^{\aleph_0}</math>, 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 curve]]s.
* A (non-empty) Hausdorff topological space is compact metrizable if and only if it is a [[continuous function (topology)|continuous]] [[image (mathematics)|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><ref>{{Cite web|url=https://imgur.com/a/UDgthQm|title=Pugh "Real Mathematical Analysis" Page 108-112 Cantor Surjection Theorem}}</ref>

Let ''C''(''X'') denote the space of all real-valued, [[bounded function|bounded]] continuous [[function (mathematics)|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|isometric]] to a [[closed set|closed]] subspace of ''C''(Δ).<ref>Carothers, ''op.cit.''</ref>
In general, this isometry is not unique, and thus is not properly a [[universal property]] in the [[category theory|categorical]] sense.

*The [[group (mathematics)|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>

== See also ==
* [[Space (mathematics)]]
* [[Cantor set]]
* [[Cantor cube]]

== References ==
<references/>

== External links ==
*{{cite book | author=Kechris, A. |authorlink = Alexander Kechris| title= Classical Descriptive Set Theory - Graduate Texts in Mathematics | url=https://archive.org/details/classicaldescrip0000kech | url-access=registration | publisher=Springer | year=1995 | isbn = 0-387-94374-9| edition=156}}

{{DEFAULTSORT:Cantor Space}}
[[Category:Topological spaces]]
[[Category:Descriptive set theory]]
[[Category:Georg Cantor]]
[[Category:Binary sequences]]