Noncompletion

Noncompletion is the state of a mathematical object that is not complete. In analysis and topology, a metric space is complete if every Cauchy sequence in the space converges to a limit that also lies in the space. A noncomplete space fails this property, meaning there exists at least one Cauchy sequence that has no limit within the space.

An archetypal example is the rational numbers Q with the usual distance. There are Cauchy sequences of rationals that converge to irrational numbers, which are not elements of Q. Therefore Q is noncomplete. The completion of Q with respect to the usual metric is the real numbers R, which contains Q densely and contains limits of all Cauchy sequences of rationals.

In general, every metric space has a completion: a larger complete space containing it as a dense subspace, unique up to isometry. Noncompletion is a property of the original space before completion; recognizing noncompletion helps explain why certain theorems that require limits or convergence do not apply unless the space is completed.

Outside pure mathematics, noncompletion can also describe processes or tasks that are not finished, but in formal contexts the term is predominantly about the absence of limits of Cauchy sequences and the resulting need for completion.