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Zero dimensional space

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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. An illustration of a nildimensional space is a point.

Contents

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.

    • References

    Zero-dimensional space Wikipedia
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