Nondiscrete

Nondiscrete describes a topological space that is not discrete. In a discrete space, every subset is open, which is equivalent to every point being isolated and every singleton being open. A nondiscrete space fails this condition: there exists at least one subset that is not open, and at least one point that is not isolated. A nondiscrete space may still contain isolated points; the presence of isolated points does not by itself imply discreteness.

Examples help illustrate the concept. The real numbers with the usual topology form a nondiscrete space, since singletons {x} are not open. The Sierpinski space, a two-point topological space with open sets ∅, {1}, and {0,1}, is also nondiscrete because {0} is not open. The cofinite topology on an infinite set—where the open sets are ∅ or those with finite complement—is another nondiscrete example, as most subsets fail to be open. By contrast, choosing the integers as a subspace of the real line yields a discrete topology, where every singleton is open.

In practice, nondiscrete spaces can be compact or non-compact, connected or totally disconnected, finite or infinite. The term emphasizes the contrast with a discrete topology and signals a richer or more subtle topological structure. See also discrete topology, topological space, and limit point for related concepts.