Zero-dimensional Space

Zero-dimensional Space

In mathematics, a zero-dimensional topological 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. 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 small inductive dimension if it has a base consisting of clopen sets.

The two notions above agree for separable, metrisable spaces.