F-6 - Zero-Dimensional Spaces

Abstract

A topological space is said to be zero-dimensional (or 0-dimensional) if it is a non-empty T1-space with a base consisting of clopen sets, that is, sets that are simultaneously closed and open. Zero-dimensional spaces form a widely studied class of topological spaces. They occur in areas as diverse as non-Archimedean analysis and the study of Boolean algebras. Also, many counterexamples in topology are zero-dimensional, though often more as a matter of convenience than intentionally. A subset of a topological space is clopen if and only if (iff) its boundary is empty. Hence, zero-dimensional spaces are exactly those having their small inductive dimension equal to zero. Obviously, zero-dimensional spaces are completely regular. A space X is strongly zero-dimensional if it is a non-empty completely regular space such that every finite cover by cozero sets has a finite refinement, which is a partition of X by clopen sets. Hence, a space X is strongly zero-dimensional iff its covering dimension is equal to zero. A topological space X is strongly zero-dimensional iff it is a nonempty completely regular space such that any two functionally separated subsets are separated by a clopen set.