Zero-dimensional space

In mathematics, a zero-dimensional topological space (or nildimensional 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. A graphical illustration of a zero-dimensional space is a point.

Definition

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 by disjoint open sets.
• A topological space is zero-dimensional with respect to the finite-to-finite covering dimension if every finite open cover of the space has a refinement that is a finite open cover 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 three notions above agree for separable, metrisable spaces.

Properties of spaces with small inductive 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 & Tkachenko 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 ${\displaystyle 2^{I}}$ where ${\displaystyle 2=\{0,1\}}$ is given the discrete topology. Such a space is sometimes called a Cantor cube. If I is countably infinite, ${\displaystyle 2^{I}}$ is the Cantor space.

Manifolds

All points of a zero-dimensional manifold are isolated.

Hypersphere

The zero-dimensional hypersphere (0-sphere) is a pair of points, and the zero-dimensional ball is a single point.

Notes

• Arhangel'skii, Alexander; Tkachenko, Mikhail (2008). Topological Groups and Related Structures. Atlantis Studies in Mathematics. Vol. 1. Atlantis Press. ISBN 978-90-78677-06-2.
• Engelking, Ryszard (1977). General Topology. PWN, Warsaw.
• Willard, Stephen (2004). General Topology. Dover Publications. ISBN 0-486-43479-6.

References