noncompleteness

Noncompleteness is the property of a mathematical structure failing to satisfy a standard completeness condition used in its area. In analysis and topology, a metric space is complete if every Cauchy sequence converges to a limit within the space. A space is noncomplete if there exists a Cauchy sequence that does not converge to any point of the space. Common examples include the rational numbers with the usual metric, where a sequence of rationals approximating sqrt(2) is Cauchy but has no rational limit, and open intervals such as (0,1) when viewed as a subset of the real line.

In order theory and lattice theory, completeness refers to the existence of certain bounds for all subsets. A poset or lattice is complete if every subset has a supremum (least upper bound) and an infimum (greatest lower bound) within the poset. If some subset lacks a supremum or infimum, the structure is noncomplete. For instance, the natural numbers under the usual order are not a complete lattice because certain infinite subsets do not have a least upper bound in N.

The concept of noncompleteness is often addressed by forming a completion, a minimal complete structure that contains the original one as a dense subset. The rational numbers complete to the real numbers, and the open interval (0,1) completes to the closed interval [0,1]. Similar completion processes exist in other mathematical settings, frequently characterized by universal properties. See also completeness, Cauchy sequence, completion, Banach space, and lattice theory.