Zero-dimensional space

In mathematics, a zero-dimensional topological space (or nildimensional) 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.^[1][2] An illustration of a nildimensional space is a point.^[3]

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 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 finite-to-finite covering dimension if every finite open cover of the space has a refinement which 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 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.

Notes

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

References

1. "zero dimensional". planetmath.org. Retrieved 2015-06-06.
2. Hazewinkel, Michiel (1989). Encyclopaedia of Mathematics, Volume 3. Kluwer Academic Publishers. p. 190.
3. Wolcott, Luke; McTernan, Elizabeth (2012). "Imagining Negative-Dimensional Space" (PDF). In Bosch, Robert; McKenna, Douglas; Sarhangi, Reza. Proceedings of Bridges 2012: Mathematics, Music, Art, Architecture, Culture. Phoenix, Arizona, USA: Tessellations Publishing. pp. 637–642. ISBN 978-1-938664-00-7. ISSN 1099-6702. Retrieved 10 July 2015.