Editing Cantor space (section)

== 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>