Complete metric space

TheInfoList

In
mathematical analysis Analysis is the branch of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical ...
, a
metric space In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gene ...
is called complete (or a Cauchy space) if every
Cauchy sequence In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
of points in has a
limit Limit or Limits may refer to: Arts and media * Limit (music), a way to characterize harmony * Limit (song), "Limit" (song), a 2016 single by Luna Sea * Limits (Paenda song), "Limits" (Paenda song), 2019 song that represented Austria in the Eurov ...
that is also in . Intuitively, a space is complete if there are no "points missing" from it (inside or at the boundary). For instance, the set of
rational number In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
s is not complete, because e.g. $\sqrt$ is "missing" from it, even though one can construct a Cauchy sequence of rational numbers that converges to it (see further examples below). It is always possible to "fill all the holes", leading to the ''completion'' of a given space, as explained below.

Definition

; Cauchy sequence : A sequence in a
metric space In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gene ...
is called Cauchy if for every positive
real number In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
there is a positive
integer An integer (from the Latin Latin (, or , ) is a classical language belonging to the Italic languages, Italic branch of the Indo-European languages. Latin was originally spoken in the area around Rome, known as Latium. Through the power of t ...
such that for all positive integers , ::. ; Expansion constant : The
expansion constant Expansion may refer to: Arts, entertainment and media * '' L'Expansion'', a French monthly business magazine * ''Expansion'' (album), by American jazz pianist Dave Burrell, released in 2004 * ''Expansions'' (McCoy Tyner album), 1970 * ''Expansio ...
of a metric space is the
infimum are equal. Image:Supremum illustration.svg, 250px, A set A of real numbers (blue circles), a set of upper bounds of A (red diamond and circles), and the smallest such upper bound, that is, the supremum of A (red diamond). In mathematics, the infim ...
of all constants $\mu$ such that whenever the family $\left\$ intersects pairwise, the intersection $\bigcap_\alpha \overline(x_\alpha, \mu r_\alpha)$ is nonempty. ; Complete space : A metric space is complete if any of the following equivalent conditions are satisfied: :#Every
Cauchy sequence In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
of points in has a
limit Limit or Limits may refer to: Arts and media * Limit (music), a way to characterize harmony * Limit (song), "Limit" (song), a 2016 single by Luna Sea * Limits (Paenda song), "Limits" (Paenda song), 2019 song that represented Austria in the Eurov ...
that is also in . :#Every Cauchy sequence in converges in (that is, to some point of ). :#The expansion constant of is ≤ 2. :#Every decreasing sequence of
non-empty#REDIRECT Empty set#REDIRECT Empty set In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, chan ...

closed
subset In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...

s of , with diameters tending to 0, has a non-empty
intersection The line (purple) in two points (red). The disk (yellow) intersects the line in the line segment between the two red points. In mathematics, the intersection of two or more objects is another, usually "smaller" object. Intuitively, the inter ...
: if is closed and non-empty, for every , and , then there is a point common to all sets .

Examples

The space Q of
rational number In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
s, with the standard metric given by the
absolute value In , the absolute value or modulus of a  , denoted , is the value of  without regard to its . Namely, if is , and if is (in which case is positive), and . For example, the absolute value of 3 is 3, and the absolute value of − ...

of the , is not complete. Consider for instance the sequence defined by and $x_ = \frac + \frac.$ 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 $x = \frac + \frac$ necessarily ''x''2 = 2, yet no rational number has this property. However, considered as a sequence of
real number In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
s, it does converge to the
irrational number In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ge ...
$\sqrt$. The
open interval In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...
, again with the absolute value 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 is complete; for example the given sequence does have a limit in this interval and the limit is zero. The space R of real numbers and the space C of
complex number In mathematics, a complex number is an element of a number system that contains the real numbers and a specific element denoted , called the imaginary unit, and satisfying the equation . Moreover, every complex number can be expressed in the for ...

s (with the metric given by the absolute value) are complete, and so is
Euclidean space Euclidean space is the fundamental space of . Originally, it was the of , but in modern there are Euclidean spaces of any nonnegative integer , including the three-dimensional space and the ''Euclidean plane'' (dimension two). It was introduce ...
R''n'', with the usual distance metric. In contrast, infinite-dimensional
normed vector space In mathematics, a normed vector space or normed space is a vector space over the Real number, real or Complex number, complex numbers, on which a Norm (mathematics), norm is defined. A norm is the formalization and the generalization to real vec ...
s may or may not be complete; those that are complete are
Banach space In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...
s. The space C 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 frame, The perimeter of the square is the set of points in R2 where the sup norm equals a fixed positive constant. In mathematical analysis, the uniform norm (or sup norm) assigns to real- or complex-valued bounded function Image:Bounded and ...
. However, the supremum norm does not give a norm on the space C of continuous functions on , for it may contain unbounded functions. Instead, with the topology of
compact convergence In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...
, C can be given the structure of a
Fréchet space In functional analysis Image:Drum vibration mode12.gif, 200px, One of the possible modes of vibration of an idealized circular drum head. These modes are eigenfunctions of a linear operator on a function space, a common construction in functional ...
: a
locally convex topological vector space In functional analysis 200px, One of the possible modes of vibration of an idealized circular drum head. These modes are eigenfunctions of a linear operator on a function space, a common construction in functional analysis. Functional analysis ...
whose topology can be induced by a complete translation-invariant metric. The space Q''p'' of ''p''-adic numbers is complete for any
prime number A prime number (or a prime) is a natural number greater than 1 that is not a Product (mathematics), product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime ...
. 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 of all
sequence In , a sequence is an enumerated collection of in which repetitions are allowed and matters. Like a , it contains (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is called the ''length'' of the sequence. Unl ...

s 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 In the field of , a homeomorphism, topological isomorphism, or bicontinuous function is a between s that has a continuous . Homeomorphisms are the s in the —that is, they are the that preserve all the of a given space. Two spaces with ...
to the product of a
countable In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
number of copies of the
discrete space In topology, a discrete space is a particularly simple example of a topological space or similar structure, one in which the points form a , meaning they are ''Isolated point, isolated'' from each other in a certain sense. The discrete topology is t ...
.
Riemannian manifold In differential geometry Differential geometry is a Mathematics, mathematical discipline that uses the techniques of differential calculus, integral calculus, linear algebra and multilinear algebra to study problems in geometry. The Differentia ...
s which are complete are called geodesic manifolds; completeness follows from the
Hopf–Rinow theorem Hopf–Rinow theorem is a set of statements about the Geodesic manifold, geodesic completeness of Riemannian manifolds. It is named after Heinz Hopf and his student Willi Rinow, who published it in 1931. Statement Let (''M'', ''g'') be a conne ...
.

Some theorems

Every
compact Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to: * Interstate compact * Blood compact, an ancient ritual of the Philippines * Compact government, a type of colonial rule utilized in British N ...
metric space is complete, though complete spaces need not be compact. In fact, a metric space is compact
if and only if In logic Logic is an interdisciplinary field which studies truth and reasoning Reason is the capacity of consciously making sense of things, applying logic Logic (from Ancient Greek, Greek: grc, wikt:λογική, λογική, l ...
it is complete and
totally boundedIn topology s, which have only one surface and one edge, are a kind of object studied in topology. In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structu ...
. This is a generalization of the Heine–Borel theorem, which states that any closed and bounded subspace of is compact and therefore complete. Let be a complete metric space. If is a closed set, then is also complete. Let be a metric space. If is a complete subspace, then is also closed. If is a Set (mathematics), set and is a complete metric space, then the set of all bounded functions from to is a complete metric space. Here we define the distance in in terms of the distance in with the
supremum norm frame, The perimeter of the square is the set of points in R2 where the sup norm equals a fixed positive constant. In mathematical analysis, the uniform norm (or sup norm) assigns to real- or complex-valued bounded function Image:Bounded and ...
:$d\left(f, g\right) \equiv \sup\left\$ If is a topological space and is a complete metric space, then the set consisting of all continuous function (topology), continuous bounded functions from to is a closed subspace of and hence also complete. The Baire category theorem says that every complete metric space is a Baire space. That is, the Union (set theory), union of Countable, countably many nowhere dense subsets of the space has empty interior (topology), interior. The Banach fixed-point theorem states that a contraction mapping on a complete metric space admits a fixed point. The fixed-point theorem is often used to prove the inverse function theorem on complete metric spaces such as Banach spaces.

Completion

For any metric space ''M'', one can 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 (mathematics), 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 ''x'' = (''x''''n'') and ''y'' = (''y''''n'') in ''M'', we may define their distance as : $d\left(x, y\right) = \lim_n d\left\left(x_n, y_n\right\right)$ (This limit exists because the real numbers are complete.) This is only a pseudometric space, 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. Georg Cantor, 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 (mathematics), 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 ''p'', 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 In mathematics, a normed vector space or normed space is a vector space over the Real number, real or Complex number, complex numbers, on which a Norm (mathematics), norm is defined. A norm is the formalization and the generalization to real vec ...
, the result is a
Banach space In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...
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

Completeness is a property of the ''metric'' and not of the ''topology'', meaning that a complete metric space can be
homeomorphic In the field of , a homeomorphism, topological isomorphism, or bicontinuous function is a between s that has a continuous . Homeomorphisms are the s in the —that is, they are the that preserve all the of a given space. Two spaces with ...
to a non-complete one. An example is given by the real numbers, which are complete but homeomorphic to the open interval , 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, Alternatives and generalizations). Indeed, some authors use the term ''topologically complete'' for a wider class of topological spaces, the completely uniformizable spaces.Kelley, Problem 6.L, p. 208 A topological space homeomorphic to a separable space, separable complete metric space is called a Polish space.

Alternatives and generalizations

Since Cauchy sequences can also be defined in general topological groups, an alternative to relying on a metric structure for defining completeness and constructing the completion of a space is to use a group structure. This is most often seen in the context of topological vector spaces, but requires only the existence of a continuous "subtraction" operation. In this setting, the distance between two points ''x'' and ''y'' is gauged not by a real number ''ε'' via the metric ''d'' in the comparison ''d''(''x'', ''y'') < ''ε'', but by an open neighbourhood ''N'' of 0 via subtraction in the comparison ''x'' − ''y'' ∈ ''N''. A common generalisation of these definitions can be found in the context of a uniform space, where an Uniform space#Entourage definition, entourage is a set of all pairs of points that are at no more than a particular "distance" from each other. It is also possible to replace Cauchy ''sequences'' in the definition of completeness by Cauchy ''net (mathematics), nets'' or filter (mathematics)#Filters in uniform spaces, Cauchy filters. If every Cauchy net (or equivalently every Cauchy filter) has a limit in ''X'', then ''X'' is called complete. One can furthermore construct a completion for an arbitrary uniform space similar to the completion of metric spaces. The most general situation in which Cauchy nets apply is Cauchy spaces; these too have a notion of completeness and completion just like uniform spaces.