Zero-dimensional space: Encyclopedia - Zero-dimensional space

Zero-dimensional space

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={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.

Adapted from the Wikipedia article "Zero-dimensional space", under the G.N U Free Docmentation License. Please also see http://en.wikipedia.org/wiki