Zero-dimensional space

Topological space of 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