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 $2^I$ 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

↑ "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.

Dimension

Dimensional spaces	Vector space Euclidean space Affine space Projective space Free module Manifold Algebraic variety Spacetime

Other dimensions	Krull Homological Lebesgue covering Inductive Hausdorff Minkowski Fractal Degrees of freedom

Polytopes and shapes	Hyperplane Hypersurface Hypercube Hypersphere Hyperrectangle Demihypercube Cross-polytope Simplex

Dimensions by number	Zero One Two Three Four Five Six (degrees of freedom) Seven Eight Nine Ten n-dimensions Negative dimensions

Category

This article is issued from Wikipedia - version of the 8/27/2016. The text is available under the Creative Commons Attribution/Share Alike but additional terms may apply for the media files.