Zero-dimensional space

Definition

• 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.

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.
• 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 is countably infinite, is the Cantor space.

  Hypersphere