Mathematical completeness
In mathematics, "complete" is a term that takes on specific meanings in specific situations, and not every situation in which a type of "completion" occurs is called a "completion". See, for example, algebraically closed field or compactification.
A metric space (or uniform space) is complete if every Cauchy sequence in it converges. See complete space.
In functional analysis, a subset S of a topological vector space V is complete if its span is dense in V. If V is separable, it follows that any vector in V can be written as a (possibly infinite) linear combination of vectors from S. In the particular case of Hilbert spaces (or more generally, inner product spaces), an orthonormal basis is a set that is both complete and orthonormal.
A measure space is complete if every subset of every null set is measurable. See complete measure.
In commutative algebra, a commutative ring can be completed at an ideal (in the topology defined by the powers of the ideal). See Completion (ring theory).
More generally, any topological group can be completed at a decreasing sequence of open subgroups.
In statistics, a statistic is called complete if it does not allow an unbiased estimator of zero. See completeness (statistics).
In graph theory, a complete graph is an undirected graph in which every pair of vertices has exactly one edge connecting them.
In category theory, a category C is complete if every diagram from a small category to C has a limit; it is cocomplete if every such functor has a colimit.
In order theory and related fields such as lattice and domain theory, completeness generally refers to the existence of certain suprema or infima of some partially ordered set. Notable special usages of the term include the concepts of complete Boolean algebra, complete lattice, and complete partial order (cpo). Furthermore, an ordered field is complete if every non-empty subset of it that has an upper bound within the field has a least upper bound within the field, which should be compared to the (slightly different) order-theoretical notion of bounded completeness. Up to isomorphism there is only one complete ordered field: the field of real numbers (but note that this complete ordered field, which is also a lattice, is not a complete lattice).
In algebraic geometry, an algebraic variety is complete if it satisfies an analog of compactness. See complete algebraic variety.
Computing
In algorithms, the notion of completeness refers to the ability of the algorithm to find a solution if one exists, and if not, reports that no solution is possible.
In computational complexity theory, a problem P is complete for a complexity class C, under a given type of reduction, if P is in C, and every problem in C reduces to P using that reduction. For example, each problem in the class NP-complete is complete for the class NP, under polynomial-time, many-one reduction.
In computing, a data-entry field can autocomplete the entered data based on the prefix typed into the field; that capability is known as autocompletion.
In software testing, completeness has for goal the functional verification of call graph (between software item) and control graph (inside each software item).
Economics, finance, and industry
Complete market
In auditing, completeness is one of the financial statement assertions that have to be ensured. For example, auditing classes of transactions. Rental expense which includes 12-month or 52-week payments should be all booked according to the terms agreed in the tenancy agreement.
Oil or gas well completion, the process of making a well ready for production.In algorithms, the notion of completeness refers to the ability of the algorithm to find a solution if one exists, and if not, reports that no solution is possible.
In computational complexity theory, a problem P is complete for a complexity class C, under a given type of reduction, if P is in C, and every problem in C reduces to P using that reduction. For example, each problem in the class NP-complete is complete for the class NP, under polynomial-time, many-one reduction.