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Concept
Zero-dimensional space
Summary
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 nildimensional space is a point.
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 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 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 where is given the discrete topology. Such a space is sometimes called a Cantor cube. If I is countably infinite, is the Cantor space.
All points of a zero-dimensional manifold are isolated.
In particular, the zero-dimensional hypersphere is a pair of points, and the zero-dimensional ball is a single point.
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