Zero-dimensional space
topological space that has small inductive dimension zero
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 (see Inductive dimension § Relationships between dimensions).
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.
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