Examples[edit]
The space Q of rational numbers, with the standard metric given by the absolute value of the difference, is not complete.
Consider for instance the sequence defined by and
This is a Cauchy sequence of rational numbers, but it does not converge towards any rational limit: If the sequence did have a limit then by solving necessarily yet no rational number has this property.
However, considered as a sequence of real numbers, it does converge to the irrational number .
The open interval (0,1), again with the absolute difference metric, is not complete either.
The sequence defined by is Cauchy, but does not have a limit in the given space.
However the closed interval [0,1] is complete; for example the given sequence does have a limit in this interval, namely zero.
The space R of real numbers and the space C of complex numbers (with the metric given by the absolute difference) are complete, and so is Euclidean space Rn, with the usual distance metric.
In contrast, infinite-dimensional normed vector spaces may or may not be complete; those that are complete are Banach spaces.
The space C[a, b] of continuous real-valued functions on a closed and bounded interval is a Banach space, and so a complete metric space, with respect to the supremum norm.
However, the supremum norm does not give a norm on the space C(a, b) of continuous functions on (a, b), for it may contain unbounded functions.
Instead, with the topology of compact convergence, C(a, b) can be given the structure of a Fréchet space: a locally convex topological vector space whose topology can be induced by a complete translation-invariant metric.
The space Qp of p-adic numbers is complete for any prime number
This space completes Q with the p-adic metric in the same way that R completes Q with the usual metric.
If is an arbitrary set, then the set SN of all sequences in becomes a complete metric space if we define the distance between the sequences and to be where is the smallest index for which is distinct from or if there is no such index.
This space is homeomorphic to the product of a countable number of copies of the discrete space
Riemannian manifolds which are complete are called geodesic manifolds; completeness follows from the Hopf–Rinow theorem.
Completion[edit]
For any metric space M, it is possible to construct a complete metric space M′ (which is also denoted as ), which contains M as a dense subspace. It has the following universal property: if N is any complete metric space and f is any uniformly continuous function from M to N, then there exists a unique uniformly continuous function f′ from M′ to N that extends f. The space M' is determined up to isometry by this property (among all complete metric spaces isometrically containing M), and is called the completion of M.
The completion of M can be constructed as a set of equivalence classes of Cauchy sequences in M. For any two Cauchy sequences and in M, we may define their distance as
(This limit exists because the real numbers are complete.) This is only a pseudometric, not yet a metric, since two different Cauchy sequences may have the distance 0. But "having distance 0" is an equivalence relation on the set of all Cauchy sequences, and the set of equivalence classes is a metric space, the completion of M. The original space is embedded in this space via the identification of an element x of M' with the equivalence class of sequences in M converging to x (i.e., the equivalence class containing the sequence with constant value x). This defines an isometry onto a dense subspace, as required. Notice, however, that this construction makes explicit use of the completeness of the real numbers, so completion of the rational numbers needs a slightly different treatment.
Cantor's construction of the real numbers is similar to the above construction; the real numbers are the completion of the rational numbers using the ordinary absolute value to measure distances. The additional subtlety to contend with is that it is not logically permissible to use the completeness of the real numbers in their own construction. Nevertheless, equivalence classes of Cauchy sequences are defined as above, and the set of equivalence classes is easily shown to be a field that has the rational numbers as a subfield. This field is complete, admits a natural total ordering, and is the unique totally ordered complete field (up to isomorphism). It is defined as the field of real numbers (see also Construction of the real numbers for more details). One way to visualize this identification with the real numbers as usually viewed is that the equivalence class consisting of those Cauchy sequences of rational numbers that "ought" to have a given real limit is identified with that real number. The truncations of the decimal expansion give just one choice of Cauchy sequence in the relevant equivalence class.
For a prime the p-adic numbers arise by completing the rational numbers with respect to a different metric.
If the earlier completion procedure is applied to a normed vector space, the result is a Banach space containing the original space as a dense subspace, and if it is applied to an inner product space, the result is a Hilbert space containing the original space as a dense subspace.
Topologically complete spaces[edit]
Completeness is a property of the metric and not of the topology, meaning that a complete metric space can be homeomorphic to a non-complete one. An example is given by the real numbers, which are complete but homeomorphic to the open interval (0,1), which is not complete.
In topology one considers completely metrizable spaces, spaces for which there exists at least one complete metric inducing the given topology. Completely metrizable spaces can be characterized as those spaces that can be written as an intersection of countably many open subsets of some complete metric space. Since the conclusion of the Baire category theorem is purely topological, it applies to these spaces as well.
Completely metrizable spaces are often called topologically complete. However, the latter term is somewhat arbitrary since metric is not the most general structure on a topological space for which one can talk about completeness (see the section Alternatives and generalizations). Indeed, some authors use the term topologically complete for a wider class of topological spaces, the completely uniformizable spaces.[3]
A topological space homeomorphic to a separable complete metric space is called a Polish space.