 Zero-dimensional space

A space is zero-dimensional when its topology has a base of clopen sets. In the Hausdorff setting these sets separate distinct points, so the space is totally disconnected. Conversely a compact Hausdorff totally disconnected space has a clopen base, by the Šura-Bura component lemma and finite subcovers of the complement of a neighborhood.

Zero-dimensional space, often denoted as \(0\)-D space, refers to a mathematical concept where a space has no dimensions. In a zero-dimensional space, all points are dimensionless, meaning there is no length, area, or volume associated with any part of the space. A classic example of a zero-dimensional space is a single point, which can be viewed as a space that contains only one element and has no extent in any direction.