Zero-dimensional space

Encyclopedia : Z : ZE : ZER : Zero-dimensional space

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In mathematics, a topological space is zero-dimensional if its topological dimension is zero, or equivalently, if it has a base consisting of clopen sets. A zero-dimensional Hausdorff space is necessarily totally disconnected, but the converse fails.

Zero-dimensional Polish spaces are a particularly convenient setting for descriptive set theory. Examples of such spaces include Cantor space and Baire space.

Hausdorff zero-dimensional spaces are precisely the subspaces of topological powers [2^I] where 2= is given the discrete topology. Such a space is sometimes called a Cantor cube. If [I] is countably infinite, [2^I] is the Cantor space.

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