complete space

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OR:

In
mathematical analysis Analysis is the branch of mathematics dealing with continuous functions, limits, and related theories, such as differentiation, integration, measure, infinite sequences, series, and analytic functions. These theories are usually studied ...
, a
metric space In mathematics, a metric space is a set together with a notion of '' distance'' between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are the most general set ...
is called complete (or a Cauchy space) if every Cauchy sequence of points in has a limit 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, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ). The set of all ratio ...
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 $x_1, x_2, x_3, \ldots$ in a
metric space In mathematics, a metric space is a set together with a notion of '' distance'' between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are the most general set ...
$\left(X, d\right)$ is called Cauchy if for every positive
real number In mathematics, a real number is a number that can be used to measure a ''continuous'' one- dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Ever ...
$r > 0$ there is a positive
integer An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign ( −1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the languag ...
$N$ such that for all positive integers $m, n > N,$ $d\left(x_m, x_n\right) < r.$ Complete space A metric space $\left(X, d\right)$ is complete if any of the following equivalent conditions are satisfied: :#Every Cauchy sequence of points in $X$ has a limit that is also in $X.$ :#Every Cauchy sequence in $X$ converges in $X$ (that is, to some point of $X$). :#Every decreasing sequence of
non-empty In mathematics, the empty set is the unique set having no elements; its size or cardinality (count of elements in a set) is zero. Some axiomatic set theories ensure that the empty set exists by including an axiom of empty set, while in othe ...
closed
subset In mathematics, set ''A'' is a subset of a set ''B'' if all elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they are unequal, then ''A'' is a proper subset o ...
s of $X,$ with diameters tending to 0, has a non-empty
intersection In mathematics, the intersection of two or more objects is another object consisting of everything that is contained in all of the objects simultaneously. For example, in Euclidean geometry, when two lines in a plane are not parallel, their ...
: if $F_n$ is closed and non-empty, $F_ \subseteq F_n$ for every $n,$ and $\operatorname\left\left(F_n\right\right) \to 0,$ then there is a point $x \in X$ common to all sets $F_n.$

# Examples

The space Q of
rational number In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ). The set of all ratio ...
s, with the standard metric given by the
absolute value In mathematics, the absolute value or modulus of a real number x, is the non-negative value without regard to its sign. Namely, , x, =x if is a positive number, and , x, =-x if x is negative (in which case negating x makes -x positive), an ...
of the difference, is not complete. Consider for instance the sequence defined by $x_1 = 1$ 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 $x,$ 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, a real number is a number that can be used to measure a ''continuous'' one- dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Ever ...
s, it does converge to the
irrational number In mathematics, the irrational numbers (from in- prefix assimilated to ir- (negative prefix, privative) + rational) are all the real numbers that are not rational numbers. That is, irrational numbers cannot be expressed as the ratio of two inte ...
$\sqrt$. The
open interval In mathematics, a (real) interval is a set of real numbers that contains all real numbers lying between any two numbers of the set. For example, the set of numbers satisfying is an interval which contains , , and all numbers in between. Othe ...
, 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 Closed may refer to: Mathematics * Closure (mathematics), a set, along with operations, for which applying those operations on members always results in a member of the set * Closed set, a set which contains all its limit points * Closed interval, ...
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 extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form ...
s (with the metric given by the absolute value) are complete, and so is
Euclidean space Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean ...
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 or complex numbers, on which a norm is defined. A norm is the formalization and the generalization to real vector spaces of the intuitive notion of "length" ...
s may or may not be complete; those that are complete are
Banach space In mathematics, more specifically in functional analysis, a Banach space (pronounced ) is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between vec ...
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 In mathematical analysis, the uniform norm (or ) assigns to real- or complex-valued bounded functions defined on a set the non-negative number :\, f\, _\infty = \, f\, _ = \sup\left\. This norm is also called the , the , the , or, when th ...
. 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 compact convergence (or uniform convergence on compact sets) is a type of convergence that generalizes the idea of uniform convergence. It is associated with the compact-open topology. Definition Let (X, \mathcal) be a topological ...
, C can be given the structure of a
Fréchet space In functional analysis and related areas of mathematics, Fréchet spaces, named after Maurice Fréchet, are special topological vector spaces. They are generalizations of Banach spaces (normed vector spaces that are complete with respect to t ...
: a
locally convex topological vector space In functional analysis and related areas of mathematics, locally convex topological vector spaces (LCTVS) or locally convex spaces are examples of topological vector spaces (TVS) that generalize normed spaces. They can be defined as topological v ...
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 ...
$p.$ This space completes Q with the ''p''-adic metric in the same way that R completes Q with the usual metric. If $S$ is an arbitrary set, then the set of all
sequence In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is call ...
s in $S$ becomes a complete metric space if we define the distance between the sequences $\left\left(x_n\right\right)$ and $\left\left(y_n\right\right)$ to be $\tfrac$ where $N$ is the smallest index for which $x_N$ is distinct from $y_N$ or $0$ if there is no such index. This space is
homeomorphic In the mathematical field of topology, a homeomorphism, topological isomorphism, or bicontinuous function is a bijective and continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isomorp ...
to the product of a
countable In mathematics, a set is countable if either it is finite or it can be made in one to one correspondence with the set of natural numbers. Equivalently, a set is ''countable'' if there exists an injective function from it into the natural numbe ...
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'' from each other in a certain sense. The discrete topology is the finest to ...
$S.$
Riemannian manifold In differential geometry, a Riemannian manifold or Riemannian space , so called after the German mathematician Bernhard Riemann, is a real, smooth manifold ''M'' equipped with a positive-definite inner product ''g'p'' on the tangent space ' ...
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 completeness of Riemannian manifolds. It is named after Heinz Hopf and his student Willi Rinow, who published it in 1931. Stefan Cohn-Vossen extended part of the Hopf–Rinow theorem ...
.

# 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 Briti ...
metric space is complete, though complete spaces need not be compact. In fact, a metric space is compact
if and only if In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false. The connective is bic ...
it is complete and
totally bounded In topology and related branches of mathematics, total-boundedness is a generalization of compactness for circumstances in which a set is not necessarily closed. A totally bounded set can be covered by finitely many subsets of every fixed “size ...
. This is a generalization of the Heine–Borel theorem, which states that any closed and bounded subspace $S$ of is compact and therefore complete. Let $\left(X, d\right)$ be a complete metric space. If $A \subseteq X$ is a closed set, then $A$ is also complete. Let $\left(X, d\right)$ be a metric space. If $A \subseteq X$ is a complete subspace, then $A$ is also closed. If $X$ is a set and $M$ is a complete metric space, then the set $B\left(X, M\right)$ of all
bounded function In mathematics, a function ''f'' defined on some set ''X'' with real or complex values is called bounded if the set of its values is bounded. In other words, there exists a real number ''M'' such that :, f(x), \le M for all ''x'' in ''X''. A ...
s from to $M$ is a complete metric space. Here we define the distance in $B\left(X, M\right)$ in terms of the distance in $M$ with the
supremum norm In mathematical analysis, the uniform norm (or ) assigns to real- or complex-valued bounded functions defined on a set the non-negative number :\, f\, _\infty = \, f\, _ = \sup\left\. This norm is also called the , the , the , or, when th ...
$d(f, g) \equiv \sup\$ If $X$ is a
topological space In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called po ...
and $M$ is a complete metric space, then the set $C_b\left(X, M\right)$ consisting of all
continuous Continuity or continuous may refer to: Mathematics * Continuity (mathematics), the opposing concept to discreteness; common examples include ** Continuous probability distribution or random variable in probability and statistics ** Continuous g ...
bounded functions $f : X \to M$ is a closed subspace of $B\left(X, M\right)$ and hence also complete. The
Baire category theorem The Baire category theorem (BCT) is an important result in general topology and functional analysis. The theorem has two forms, each of which gives sufficient conditions for a topological space to be a Baire space (a topological space such that t ...
says that every complete metric space is a
Baire space In mathematics, a topological space X is said to be a Baire space if countable unions of closed sets with empty interior also have empty interior. According to the Baire category theorem, compact Hausdorff spaces and complete metric spaces are e ...
. That is, the union of
countably many In mathematics, a set is countable if either it is finite or it can be made in one to one correspondence with the set of natural numbers. Equivalently, a set is ''countable'' if there exists an injective function from it into the natural numbers; ...
nowhere dense In mathematics, a subset of a topological space is called nowhere dense or rare if its closure has empty interior. In a very loose sense, it is a set whose elements are not tightly clustered (as defined by the topology on the space) anywhere. ...
subsets of the space has empty
interior Interior may refer to: Arts and media * ''Interior'' (Degas) (also known as ''The Rape''), painting by Edgar Degas * ''Interior'' (play), 1895 play by Belgian playwright Maurice Maeterlinck * ''The Interior'' (novel), by Lisa See * Interior de ...
. The
Banach fixed-point theorem In mathematics, the Banach fixed-point theorem (also known as the contraction mapping theorem or contractive mapping theorem) is an important tool in the theory of metric spaces; it guarantees the existence and uniqueness of fixed points of cert ...
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 In mathematics, specifically differential calculus, the inverse function theorem gives a sufficient condition for a function to be invertible in a neighborhood of a point in its domain: namely, that its ''derivative is continuous and non-zero at t ...
on complete metric spaces such as Banach spaces.

# Completion

For any metric space ''M'', it is possible to construct a complete metric space ''M′'' (which is also denoted as $\overline$), 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 In mathematics, an isometry (or congruence, or congruent transformation) is a distance-preserving transformation between metric spaces, usually assumed to be bijective. The word isometry is derived from the Ancient Greek: ἴσος ''isos'' mea ...
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 class In mathematics, when the elements of some set S have a notion of equivalence (formalized as an equivalence relation), then one may naturally split the set S into equivalence classes. These equivalence classes are constructed so that elements a ...
es of Cauchy sequences in ''M''. For any two Cauchy sequences $x_ = \left\left(x_n\right\right)$ and $y_ = \left\left(y_n\right\right)$ in ''M'', we may define their distance as $d\left(x_, y_\right) = \lim_n d\left(x_n, y_n\right)$ (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 In mathematics, an equivalence relation is a binary relation that is reflexive, symmetric and transitive. The equipollence relation between line segments in geometry is a common example of an equivalence relation. Each equivalence relatio ...
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 In mathematics, an isometry (or congruence, or congruent transformation) is a distance-preserving transformation between metric spaces, usually assumed to be bijective. The word isometry is derived from the Ancient Greek: ἴσος ''isos'' mea ...
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 Field may refer to: Expanses of open ground * Field (agriculture), an area of land used for agricultural purposes * Airfield, an aerodrome that lacks the infrastructure of an airport * Battlefield * Lawn, an area of mowed grass * Meadow, a grass ...
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 In mathematics, there are several equivalent ways of defining the real numbers. One of them is that they form a complete ordered field that does not contain any smaller complete ordered field. Such a definition does not prove that such a complete ...
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 -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 or complex numbers, on which a norm is defined. A norm is the formalization and the generalization to real vector spaces of the intuitive notion of "length" ...
, the result is a
Banach space In mathematics, more specifically in functional analysis, a Banach space (pronounced ) is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between vec ...
containing the original space as a dense subspace, and if it is applied to an
inner product space In mathematics, an inner product space (or, rarely, a Hausdorff pre-Hilbert space) is a real vector space or a complex vector space with an operation called an inner product. The inner product of two vectors in the space is a scalar, often de ...
, the result is a
Hilbert space In mathematics, Hilbert spaces (named after David Hilbert) allow generalizing the methods of linear algebra and calculus from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise natural ...
containing the original space as a dense subspace.

# Topologically complete spaces

Completeness is a property of the ''metric'' and not of the ''
topology In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing ...
'', meaning that a complete metric space can be
homeomorphic In the mathematical field of topology, a homeomorphism, topological isomorphism, or bicontinuous function is a bijective and continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isomorp ...
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 In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing ...
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 The Baire category theorem (BCT) is an important result in general topology and functional analysis. The theorem has two forms, each of which gives sufficient conditions for a topological space to be a Baire space (a topological space such that t ...
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 space In mathematics, a topological space (''X'', ''T'') is called completely uniformizable (or Dieudonné complete) if there exists at least one complete uniformity that induces the topology ''T''. Some authors additionally require ''X'' to be Hausdorf ...
s.Kelley, Problem 6.L, p. 208 A topological space homeomorphic to a separable complete metric space is called a
Polish space In the mathematical discipline of general topology, a Polish space is a separable completely metrizable topological space; that is, a space homeomorphic to a complete metric space that has a countable dense subset. Polish spaces are so named b ...
.

# Alternatives and generalizations

Since Cauchy sequences can also be defined in general
topological group In mathematics, topological groups are logically the combination of groups and topological spaces, i.e. they are groups and topological spaces at the same time, such that the continuity condition for the group operations connects these two st ...
s, 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 space In mathematics, a topological vector space (also called a linear topological space and commonly abbreviated TVS or t.v.s.) is one of the basic structures investigated in functional analysis. A topological vector space is a vector space that is als ...
s, 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 $\varepsilon$ via the metric $d$ in the comparison $d\left(x, y\right) < \varepsilon,$ but by an open neighbourhood $N$ of $0$ via subtraction in the comparison $x - y \in N.$ A common generalisation of these definitions can be found in the context of a
uniform space In the mathematical field of topology, a uniform space is a set with a uniform structure. Uniform spaces are topological spaces with additional structure that is used to define uniform properties such as completeness, uniform continuity and un ...
, where an 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 '' nets'' or 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 space In general topology and analysis, a Cauchy space is a generalization of metric spaces and uniform spaces for which the notion of Cauchy convergence still makes sense. Cauchy spaces were introduced by H. H. Keller in 1968, as an axiomatic tool deriv ...
s; these too have a notion of completeness and completion just like uniform spaces.