Cantor space

From Academic Kids

In mathematics, the term Cantor space 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.

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 2N or 2ω (where 2 denotes the 2-element set {0,1} with the discrete topology). A point in 2N 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

<math>

\sum_{n=1}^\infty \frac{2 a_n}{3^n}. <math>

It is not difficult to see that this mapping is a homeomorphism from 2N onto the Cantor set, and hence that 2N is indeed a Cantor space.

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".) This 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.

It 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 it is usually easiest to deal with 2N, 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 <math>2^{\aleph_0}<math>, 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.

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