noncomplete

Noncomplete is a mathematical term used to describe a metric or uniform space in which not all Cauchy sequences converge to a point within the space. In a complete space, every Cauchy sequence has a limit inside the space; noncomplete spaces fail this property.

In relation to completion, every metric space has a unique completion, up to isometry, that is a complete space containing the original space as a dense subset. The completion can be constructed by forming equivalence classes of Cauchy sequences in the original space or by embedding the space into a larger complete space and taking closures. The completion provides a canonical setting in which limit processes can always be carried out.

Common examples of noncomplete spaces include the rational numbers with the usual metric, where Cauchy sequences may converge to irrational numbers not contained in Q; and the open interval (0,1) with the standard metric, where sequences such as 1/n are Cauchy but converge to 0, which lies outside the interval. In contrast, the real numbers R, any closed subspace of a complete space, and any Banach space are complete. Open subsets of complete spaces are frequently noncomplete, illustrating how completion depends on the ambient space.

Understanding noncomplete spaces is important because many naturally arising spaces in analysis and topology are noncomplete, and passing to their completions allows the use of convergence arguments and the application of powerful theorems that require completeness.