The Cantor set itself is of course a Cantor space. But the canonical example of a Cantor space is the countably infinite topological product of the discrete 2-point space . This is usually denoted by
or by(where denotes the 2-element set
with the discrete topology).A point in is an infinte binary sequence, that is a sequence which assumes only the values 0 or 1. Given such a sequence , one can map it to the real number
.
It is not difficult to see that this mapping is a homeomorphism from onto the Cantor set, and hence that
is a Cantor space.A topological characterization of Cantor spaces is given by Brouwer's theorem: Any two compact Hausdorff spaces without isolated points and having countable bases consisiting of clopen sets are homeomorphic to each other. The theorem can also be restated as: A topological space is a Cantor space if and only if it is a perfect, compact, totally disconnected, metrizable space. It is also equivalent (via Stone's duality) to the fact that any two countable atomless Boolean algebras are isomorphic.
The topological property of having a base consisting of clopen sets is sometimes known as "zero-dimesionality."
As can be expected from Brouwer's theorem, Cantor spaces appear in several forms. But it is usually easiest to deal with , since because of its special product form, many topological and other properties are brought out very explicitly.
For example, it becomes obvious that the cardinality of any Cantor space is , that is, the cardinality of the continuum. Also clear is the fact that the product of two (or even any finite or countable number of) Cantor spaces is a Cantor space - an important fact about Cantor spaces.
Using this last fact and the Cantor function, it is easy to construct Space-filling curves.
Cantor spaces occur in abundance 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 construction of the usual Cantor set.) Also, every uncountable, separable, completely metrizable space contains Cantor spaces as subspaces. This includes most of the common type of spaces in real analysis. As a corollary, we see that separable, completely metrizable spaces satisfy the Continuum hypothesis: Every such space is either countable or has the cardinality of the continuum.
Compact metric spaces are also closely related to Cantor spaces: A Hausdorff topological space is compact metrizable if and only if it is a continuous image of a Cantor space.