Inmathematics, azero-dimensional topological space (ornildimensional space) is atopological space that has dimension zero with respect to one of several inequivalent notions of assigning adimension to a given topological space.^[1] A graphical illustration of a zero-dimensional space is apoint.^[2]

Specifically:

• A topological space is zero-dimensional with respect to theLebesgue covering dimension if everyopen cover of the space has arefinement 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 thesmall inductive dimension if it has abase consisting ofclopen sets.

The three notions above agree forseparable,metrisable spaces (seeInductive dimension § Relationships between dimensions).

• A zero-dimensionalHausdorff space is necessarilytotally disconnected, but the converse fails. However, alocally 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-dimensionalPolish spaces are a particularly convenient setting fordescriptive set theory. Examples of such spaces include theCantor space andBaire space.
• Hausdorff zero-dimensional spaces are precisely thesubspaces of topologicalpowers${\displaystyle 2^I}$ where${\displaystyle 2=\{0,1\}}$ is given thediscrete topology. Such a space is sometimes called aCantor cube. IfI iscountably infinite,${\displaystyle 2^I}$ is the Cantor space.

All points of a zero-dimensionalmanifold areisolated.

• 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.

• Dimension
• Dimension theory
• Descriptive set theory
• Properties of topological spaces
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