Zero-dimensional space

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This article is about zero dimension in topology. For several kinds of zero space in algebra, see zero object (algebra).

In mathematics, a zero-dimensional topological space is a topological space that has dimension zero with respect to one of several inequivalent notions of assigning a dimension to a given topological space. Specifically:

The two notions above agree for separable, metrisable spaces.

Properties of spaces with covering dimension zero[edit]

A zero-dimensional Hausdorff space is necessarily totally disconnected, but the converse fails. However a locally compact Hausdorff space is zero-dimensional if and only if it is totally disconnected. (See (Arhangel'skii 2008, Proposition 3.1.7, p.136) for the non-trivial direction.)

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

Hausdorff zero-dimensional spaces are precisely the subspaces of topological powers 2^I where 2={0,1} 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.

References[edit]