# Zero-dimensional space

This article is about zero dimension in topology. For several kinds of zero space in algebra, see zero object (algebra).

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 where is given the discrete topology. Such a space is sometimes called a Cantor cube. If I is countably infinite, 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

- ↑ "zero dimensional". planetmath.org. Retrieved 2015-06-06.
- ↑ Hazewinkel, Michiel (1989).
*Encyclopaedia of Mathematics, Volume 3*. Kluwer Academic Publishers. p. 190. - ↑ 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.

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