Zero-dimensional space

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In mathematics, a topological space is zero-dimensional or 0-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. However a locally compact Hausdorff space is zero-dimensional if and only if it is totally disconnected.

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.

[edit] References

• R. Engelking (1977). General Topology. PWN, Warsaw. 
• Wilard, Stephen (2004). General Topology. Dover Publications. ISBN 0-486-43479-6.