Zero-dimensional space

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 (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

• Arhangel'skii, Alexander (2008), Topological groups and related structures, Atlantis studies in mathematics, Vol. 1, Atlantis Press, ISBN 9078677066 
• Engelking, Ryszard (1977). General Topology. PWN, Warsaw. 
• Willard, Stephen (2004). General Topology. Dover Publications. ISBN 0-486-43479-6.