For questions related to zero dimensionality. It's a topological space having a base of sets that are at the same time open and closed in it.
A zero-dimensional topological space (or nildimensional space) is a topological space that has dimension $0$ 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 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.