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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 Space s (or more generally, Inner Product Space s), 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 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 Functor from a small category to ''C'' has a Limit ; it is ''cocomplete'' if every such functor has a Colimit . For more information, see the given article on limits in category theory.

In s.

In Logic , a formal Calculus (often just specified by a set of additional Axiom s used to formalize some theory within the underlying logic) is ''complete'' if, for any statement ''P'', if $P$ is true in every Model of the theory, then a proof exists for ''P''. First Order Logic is known to be complete. A Theory is ''complete'' if it contains either $S$ or

S

Gödel's Incompleteness Theorem