• Source: Zero-dimensional space

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. A graphical illustration of a zero-dimensional space is a point.




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 (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.




Manifolds


All points of a zero-dimensional manifold are isolated.




Notes


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




References

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