In mathematics, a Cantor space, named for Georg Cantor, is a 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 2ω is called "the" Cantor space.
Cantor spaces occur abundantly in real analysis. For example, they exist as subspaces in every perfect, 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 constructionof the usual Cantor set.) Also, every uncountable, separable, completely metrizable space containsCantor spaces as subspaces. This includes most of the common type of spaces in real analysis.
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-dimensionality". 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.
As can be expected from Brouwer's theorem, Cantor spaces appear in several forms. But many properties of Cantor spaces can be established using 2ω, because its construction as a product makes it amenable to analysis.
Cantor spaces have the following properties:
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: