Zerodimensional

Zerodimensional, or zero-dimensional, describes a property of a topological space in which its topology has a base consisting entirely of clopen sets (sets that are both open and closed). A basis of clopen sets implies that around every point you can find neighborhoods that are simultaneously open and closed and refine any given neighborhood.

Equivalently, a zerodimensional space is typically understood as being totally disconnected, meaning its connected subsets are singletons. In many contexts, zerodimensionality is taken to mean the existence of a base of clopen sets, which immediately yields a wealth of separations by clopen neighborhoods: for any point and any closed set not containing it, there is a clopen neighborhood of the point disjoint from the closed set.

In metric spaces, zerodimensionality is characterized by the presence of a base consisting of clopen sets. Classic examples include the Cantor set and the space of p-adic integers; discrete spaces (where every subset is open and closed) are trivially zerodimensional. By contrast, the real line with its standard topology is not zerodimensional, since its only clopen sets are the empty set and the whole line.

In the realm of dimension theory, zerodimensionality is related to the notion of topological dimension. For well-behaved spaces such as compact Hausdorff spaces, zero-dimensionality is equivalent to being a Stone space: a compact, totally disconnected, zero-dimensional space whose clopen sets form a base. Stone’s representation theorem connects such spaces to Boolean algebras of clopen sets, a viewpoint central to logic and topology.

The term is commonly written as zero-dimensional or zerodimensional, and is used across topology, analysis, and related fields to distinguish spaces of dimension zero from those of higher dimensions.