Cantor Space - Properties

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 2ω, because its construction as a product makes it amenable to analysis.

Cantor spaces have the following properties:

• The cardinality of any Cantor space is, 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.

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:

• C(K) is isometric to a closed subspace of C(Δ).

In general, this isometry is not unique, and thus is not properly a universal property in the categorical sense.

• The group of all homeomorphisms of the Cantor space is simple.