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

• Definition

• Properties of spaces with small inductive dimension zero

• Manifolds

• Notes

• References

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.

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

All points of a zero-dimensional manifold are isolated.

• citerefarhangel-skiitkachenko2008Arhangel'skii, Alexander; Tkachenko, Mikhail (2008). Topological Groups and Related Structures. Atlantis Studies in Mathematics. Vol. 1. Atlantis Press. ISBN 978-90-78677-06-2.

• citerefengelking-ryszard1977Engelking, Ryszard (1977). General Topology. PWN, Warsaw.

• citerefwillard-stephen2004Willard, Stephen (2004). General Topology. Dover Publications. ISBN 0-486-43479-6.

cite-note-11. ↑ citerefhazewinkel1989Hazewinkel, Michiel (1989). Encyclopaedia of Mathematics, Volume 3. Kluwer Academic Publishers. p. 190. ISBN 9789400959941.

cite-note-22. ↑ citerefwolcottmcternan2012Wolcott, Luke; McTernan, Elizabeth (2012). "Imagining Negative-Dimensional Space" (PDF). In Bosch, Robert; McKenna, Douglas; Sarhangi, Reza (eds.). 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. Archived from the original (PDF) on 26 June 2015. Retrieved 10 July 2015.