Completeness - Mathematical Completeness

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.

• The completeness of the real numbers is one of the defining properties of the real number system. It may be described equivalently as either the completeness of R as metric space or as a partially ordered set (see below).

• A metric space is complete if every Cauchy sequence in it converges. See Complete metric space.

• A uniform space is complete if every Cauchy net in it converges (or equivalently every Cauchy filter in it converges).

• In functional analysis, a subset S of a topological vector space V is complete if its span is dense in V. 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.

• In quantum mechanics, a complete set of commuting operators (or CSCO) is one whose eigenvalues are sufficient to specify the physical state of a system.