# Zero-dimensional space

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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:

- A topological space is zero-dimensional with respect to the Lebesgue covering dimension if every open cover of the space has a refinement which is a cover of the space by open sets such that any point in the space is contained in exactly one open set of this refinement.
- A topological space is zero-dimensional with respect to the small inductive dimension if it has a base consisting of clopen sets.

The two notions above agree for separable, metrisable spaces.

## Properties of spaces with covering dimension zero

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 Template:Harv 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 where 2={0,1} is given the discrete topology. Such a space is sometimes called a Cantor cube. If Template:Mvar is countably infinite, is the Cantor space.

## References

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