Editing Cantor space (section)

== 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.