Cantor topology

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 2^ω is called "the" Cantor space. Note that, commonly, 2^ω is referred to simply as the Cantor set, while the term Cantor space is reserved for the more general construction of D^S for a finite set D and a set S which might be finite, countable or possibly uncountable.

The Cantor set itself is 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 ${\displaystyle 2^{\mathbb {N} }}$ 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 a₀, a₁, a₂,..., one can map it to the real number

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

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