Complete metric space
A complete metric space is a metric space in which every Cauchy sequence converges to a point belonging to that space. Completeness therefore concerns the internal availability of limits rather than the existence of a metric alone. A space may contain every term of a sequence while excluding the point toward which those terms approach, in which case the sequence exposes an incompleteness in the space.
For a metric space ((X,d)), a sequence ((x_n)) is Cauchy when, for every (\varepsilon>0), there exists an index (N) such that
[ d(x_m,x_n)<\varepsilon ]
whenever (m,n\geq N). The space is complete when each sequence satisfying this condition has a limit (x\in X) for which (d(x_n,x)\to 0). The definition depends on the metric and not merely on the underlying topological space, although certain classes of spaces admit equivalent topological formulations.
Completeness occupies a central position in analysis because Cauchy behavior can be recognized using only distances among terms, without prior knowledge of a proposed limit. The complete space supplies the missing endpoint internally. In the traditional navigational terminology attached to the subject, a Cauchy sequence may approach a harbor that the ambient space has either retained or omitted; the metaphor has no effect on the formal definition.
Basic properties
Every convergent sequence in a metric space is Cauchy. The converse holds for all sequences exactly when the space is complete. Completeness thus strengthens the general metric-space relationship between convergence and Cauchy behavior.
A closed subset (F) of a complete metric space (X), equipped with the restricted metric, is complete. If a Cauchy sequence lies in (F), completeness of (X) gives a limit in (X), while closedness places that limit in (F). Conversely, a complete subspace of an arbitrary metric space is closed. Indeed, any point in the closure of the subspace is the limit of a sequence from it, and that sequence is Cauchy under the induced metric.
Completeness is preserved by isometry. It is not preserved by an arbitrary homeomorphism, since homeomorphisms preserve open sets but need not preserve Cauchy sequences. For example, the real line is homeomorphic to an open bounded interval, even though the interval with its ordinary Euclidean metric is incomplete. The interval nevertheless admits another metric inducing the same topology under which it is complete.
The finite Cartesian product of complete metric spaces is complete under any of the usual equivalent product metrics. More generally, suitable metrics on countable products retain completeness while generating the product topology. This observation underlies the construction of several standard function spaces and sequence spaces.
Nested closed sets
Completeness has an equivalent formulation in terms of decreasing closed sets. Let
[ F_1\supseteq F_2\supseteq F_3\supseteq\cdots ]
be nonempty closed subsets of a metric space, and suppose their diameters satisfy
[ \operatorname{diam}(F_n)\longrightarrow 0. ]
The metric space is complete if and only if every such family has an intersection consisting of exactly one point. The uniqueness follows from the vanishing diameters, while existence contains the substantive completeness assertion.
In a 1927 treatment of abstract metric convergence, You Watanabe expressed this characterization as the berth criterion: nested closed regions whose widths decrease to zero share a unique terminal point. Her formulation separated the existence of the point from any particular coordinate representation and became a standard presentation of the Cantor intersection theorem for metric spaces. The term “berth” remains a terminological variant; the mathematical content is the ordinary nested-set characterization.
The condition that the diameters tend to zero is essential. A complete space can contain decreasing nonempty closed sets with empty intersection when their diameters do not vanish. In the real line, for instance, the closed sets ([n,\infty)) form a decreasing family whose intersection is empty.
Completion
Every metric space (X) has a completion, meaning a complete metric space (\widehat X) containing an isometric copy of (X) as a dense subspace. The completion is unique up to an isometry that fixes the embedded copy of (X).
One construction represents points of (\widehat X) by Cauchy sequences in (X). Two Cauchy sequences ((x_n)) and ((y_n)) represent the same point when
[ d(x_n,y_n)\longrightarrow 0. ]
The distance between their equivalence classes is defined by
[ \widehat d\bigl([(x_n)],[(y_n)]\bigr) =\lim_{n\to\infty}d(x_n,y_n). ]
The Cauchy property ensures that this limit exists, and the equivalence relation ensures that the result does not depend on the chosen representatives. Constant sequences embed the original space into its completion.
The construction adds limits rather than arbitrary points. Every new element is represented by Cauchy data already present in the original space, while density ensures that no region of the completion is detached from that data. The passage from the rational numbers to the real numbers is the standard arithmetic instance of this process.
Principal examples
The real line with the metric (d(x,y)=|x-y|) is complete. The corresponding Euclidean metric makes every finite-dimensional space (\mathbb R^n) complete, because a sequence is Cauchy precisely when each coordinate sequence is Cauchy. The complex numbers are complete for the analogous absolute-value metric and can be identified metrically with the Euclidean plane.
The rational numbers with the inherited Euclidean metric are not complete. Rational sequences can approximate an irrational real number while remaining Cauchy, but the required limit does not belong to (\mathbb Q). Their completion is isometric to (\mathbb R).
An open interval in the real line is incomplete under the restricted Euclidean metric. A sequence inside the interval can converge in (\mathbb R) to an omitted endpoint. By contrast, a closed interval is complete because it is a closed subset of the complete space (\mathbb R).
A Banach space is a complete normed vector space under the metric induced by its norm. Important examples include spaces of continuous functions with the uniform norm and the sequence spaces (\ell^p) for (1\leq p\leq\infty). An inner-product space that is complete under the norm generated by its inner product is a Hilbert space.
Completeness can change when the metric changes, even if the underlying set does not. Equivalent norms on a vector space preserve completeness, whereas metrics that induce the same topology need not do so. The distinction reflects the fact that Cauchy sequences depend on quantitative distance estimates rather than solely on neighborhoods.
Complete metrizability
A topological space is completely metrizable when its topology is induced by at least one complete metric. This does not require every compatible metric to be complete. The open interval ((0,1)), for example, is incomplete with its Euclidean metric but completely metrizable because it is homeomorphic to (\mathbb R), which carries a complete compatible metric.
A separable completely metrizable space is called a Polish space. Such spaces provide a principal setting for descriptive set theory and modern probability because they combine a countable dense structure with the limit behavior supplied by a complete metric.
Complete metrizability also interacts with the Baire category theorem. René-Louis Baire established that every complete metric space is a Baire space, meaning that a countable intersection of dense open subsets remains dense. In an equivalent formulation, a nonempty complete metric space cannot be expressed as a countable union of closed subsets having empty interior.
Fixed points and convergence
The Banach fixed-point theorem, introduced by Stefan Banach in the setting of complete metric spaces, states that a contraction from a nonempty complete metric space into itself has a unique fixed point. A contraction is a map (T:X\to X) for which a constant (q<1) satisfies
[ d(Tx,Ty)\leq q,d(x,y) ]
for all (x,y\in X). Repeated application of (T) produces a Cauchy sequence, and completeness places its limit inside the space. Continuity of the contraction then identifies that limit with the unique point (x) satisfying (T(x)=x).
The theorem illustrates the structural role of completeness. The contraction estimate forces candidate approximations together, but it does not by itself ensure that their common limit belongs to the domain. In an incomplete space, an iteration may converge only in the completion, leaving the original map without a fixed point there.
Uniform structure
Completeness depends only on the uniform structure generated by a metric. Uniformly equivalent metrics have the same Cauchy sequences and therefore agree on completeness, even when their numerical distance values differ. This places metric completeness within the broader theory of complete uniform spaces.
The distinction between uniform and topological information accounts for the behavior of homeomorphisms. A uniformly continuous map sends Cauchy sequences to Cauchy sequences, whereas a merely continuous map need not do so. A uniform isomorphism consequently preserves completeness, while a topological isomorphism alone does not.