Cantor space

In mathematics, the term Cantor space (named for Georg Cantor) is sometimes used to denote the 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^ω is called "the" Cantor space.

Examples

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 {0, 1}. This is usually written as $2^{\mathbb {N} }$ or 2^ω (where 2 denotes the 2-element set {0,1} with the discrete topology). A point in 2^ω is an infinite binary sequence, that is a sequence which assumes only the values 0 or 1. Given such a sequence a₁, a₂, a₃,..., one can map it to the real number

$\sum _{n=1}^{\infty }{\frac {2a_{n}}{3^{n}}}.$

This mapping gives is a homeomorphism from 2^ω onto the Cantor set, demonstrating that 2^ω is indeed a Cantor space.

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.

Characterization

A topological characterization of Cantor spaces is given by Brouwer's theorem:

Any two non-empty compact Hausdorff spaces without isolated points and having countable bases consisting of clopen sets are homeomorphic to each other.

The topological property of having a base consisting of clopen sets is sometimes known as "zero-dimensionality". Brouwer's theorem can be restated as:

A topological space is a Cantor space if and only if it is non-empty, 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 algebras 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^ω, because its construction as a product makes it amenable to analysis.

Cantor spaces have the following properties:

• The cardinality of any Cantor space is $2^{\aleph _{0}}$, 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 curves.
• A Hausdorff topological space is compact metrizable if and only if it is a continuous image of a Cantor space.

See also

• Cantor cube
• Georg Cantor

References

• Kechris, A. (1995). Classical Descriptive Set Theory (Graduate Texts in Mathematics 156 ed.). Springer. ISBN 0-387-94374-9.