Zero-dimensional space explained

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.^[1] A graphical illustration of a zero-dimensional space is a point.^[2]

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 that 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 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.

2 ^{I} where 2= \{ 0 , 1 \} is given the discrete topology. Such a space is sometimes called a Cantor cube. If is countably infinite, 2 ^{I} is the Cantor space.

Manifolds

All points of a zero-dimensional manifold are isolated.

Notes

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

Notes and References

1. Book: Hazewinkel, Michiel. Encyclopaedia of Mathematics, Volume 3. 1989. Kluwer Academic Publishers. 190. 9789400959941.
2. Luke. Wolcott. Elizabeth. McTernan. Imagining Negative-Dimensional Space. 637–642. Proceedings of Bridges 2012: Mathematics, Music, Art, Architecture, Culture. 2012. Robert. Bosch. Douglas. McKenna. Reza. Sarhangi. 978-1-938664-00-7. 1099-6702. Tessellations Publishing. Phoenix, Arizona, USA. 10 July 2015.