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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 2ω is called "the" Cantor space. Note that, commonly, 2ω 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.〔Stephen Willard, ''General Topology'' (1970) Addison-Wesley Publishing. ''See section 17.9a''〕 == Examples == 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 . This is usually written as or 2ω (where 2 denotes the 2-element set 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 ''a''0, ''a''1, ''a''2,..., 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 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. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Cantor space」の詳細全文を読む スポンサード リンク
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