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 or 2ω (where 2 denotes the 2-element set {0,1} with the discrete topology). A point in 2ω 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
This mapping gives a homeomorphism from 2ω onto the Cantor set, demonstrating that 2ω 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.
Read more about this topic: Cantor Space
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