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A metric space in which every Cauchy sequence converges to a point of the space.
A complete metric space is a metric space in which every Cauchy sequence converges (as a convergent sequence) to a point of .
Completeness is central in analysis and topology; for example it interacts strongly with compactness (see compact iff complete and totally bounded) and with the Baire category theorem (see complete metric spaces are Baire).
A Cauchy sequence in a metric space is a sequence such that for every there exists with
A convergent sequence in a topological space converges to a point if for every neighborhood of , there exists such that for all .
In a metric space , this is equivalent to . In a Hausdorff space, limits of convergent sequences are unique (see uniqueness of limits).
A compact set is a subset of a topological space such that every open cover of contains a finite subcover: whenever is a family of open sets with , there exist indices such that
Compact iff complete and totally bounded: Let be a metric space. Then the following are equivalent:
Baire category theorem: Let be a complete metric space. If is a sequence of dense open sets in , then the intersection is dense in .
Complete metric space is Baire: Every complete metric space is a Baire space; equivalently, in a complete metric space, every countable intersection of dense open sets is dense.