Editing Complete metric space

{{Short description|Metric geometry}}
{{Redirect|Cauchy completion|the use in category theory|Karoubi envelope}}
In [[mathematical analysis]], a [[metric space]] {{mvar|M}} is called '''complete''' (or a '''Cauchy space''') if every [[Cauchy sequence#In a metric space|Cauchy sequence]] of points in {{mvar|M}} has a [[Limit of a sequence#Metric spaces|limit]] that is also in {{mvar|M}}.

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]]s is not complete, because e.g. [[Square root of 2|<math>\sqrt{2}</math>]] 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]] <math>x_1, x_2, x_3, \ldots</math> of elements from <math>X</math> of a [[metric space]] <math>(X, d)</math> is called '''Cauchy''' if for every positive [[real number]] <math>r > 0</math> there is a positive [[integer]] <math>N</math> such that for all positive integers <math>m, n > N,</math> <math display=block>d(x_m, x_n) < r.</math> 

'''Complete space'''

A metric space <math>(X, d)</math> is '''complete''' if any of the following equivalent conditions are satisfied:
#Every Cauchy sequence in <math>X</math> converges in <math>X</math> (that is, has a limit that is also in <math>X</math>).
#Every decreasing sequence of [[empty set|non-empty]] [[closed subset]]s of <math>X,</math> with [[Diameter of a set|diameters]] tending to&nbsp;0, has a non-empty [[Intersection (set theory)|intersection]]: if <math>F_n</math> is closed and non-empty, <math>F_{n+1} \subseteq F_n</math> for every <math>n,</math> and <math>\operatorname{diam}\left(F_n\right) \to 0,</math> then there is a unique point <math>x \in X</math> common to all sets <math>F_n.</math>

==Examples==

The space <math>\Q</math> of rational numbers, with the standard [[metric (mathematics)|metric]] given by the [[absolute value]] of the [[subtraction|difference]], is not complete. 
Consider for instance the sequence defined by 
:<math>x_1 = 1\;</math> and <math>\;x_{n+1} = \frac{x_n}{2} + \frac{1}{x_n}.</math> 
This is a Cauchy sequence of rational numbers, but it does not converge towards any rational limit: If the sequence did have a limit <math>x,</math> then by solving <math>x = \frac{x}{2} + \frac{1}{x}</math> necessarily <math>x^2 = 2,</math> yet no rational number has this property. 
However, considered as a sequence of [[real number]]s, it does converge to the [[irrational number]] <math>\sqrt{2}</math>.

The [[open interval]] {{open-open|0,1}}, again with the absolute difference metric, is not complete either. 
The sequence defined by <math>x_n = \tfrac{1}{n}</math> is Cauchy, but does not have a limit in the given space. 
However the [[closed interval]] [[unit interval|{{closed-closed|0,1}}]] is complete; for example the given sequence does have a limit in this interval, namely zero.

The space <math>\R</math> of real numbers and the space <math>\C</math> of [[complex number]]s (with the metric given by the absolute difference) are complete, and so is [[Euclidean space]] <math>\R^n</math>, with the [[Euclidean distance|usual distance]] metric. 
In contrast, [[dimension (vector space)|infinite-dimensional]] [[normed vector space]]s may or may not be complete; those that are complete are [[Banach space]]s. 
The space C{{closed-closed|''a'', ''b''}} of [[continuous functions on a compact Hausdorff space|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{{open-open|''a'', ''b''}} of continuous functions on {{open-open|''a'', ''b''}}, for it may contain [[bounded function|unbounded functions]]. 
Instead, with the [[topological space|topology]] of [[compact convergence]], C{{open-open|''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 [[Metric space#Normed vector spaces|translation-invariant]] metric.

The space '''Q'''<sub>''p''</sub> of [[p-adic number|''p''-adic numbers]] is complete for any [[prime number]] <math>p.</math> 
This space completes '''Q''' with the ''p''-adic metric in the same way that '''R''' completes '''Q''' with the usual metric.

If <math>S</math> is an arbitrary set, then the set {{math|''S''<sup>'''N'''</sup>}} of all sequences in <math>S</math> becomes a complete metric space if we define the distance between the sequences <math>\left(x_n\right)</math> and <math>\left(y_n\right)</math> to be <math>\tfrac{1}{N}</math> where <math>N</math> is the smallest index for which <math>x_N</math> is [[Distinct (mathematics)|distinct]] from <math>y_N</math> or <math>0</math> if there is no such index. 
This space is [[homeomorphic]] to the [[product topology|product]] of a [[countable]] number of copies of the [[discrete space]] <math>S.</math>

[[Riemannian manifold]]s which are complete are called [[geodesic manifold]]s; completeness follows from the [[Hopf–Rinow theorem]].

==Some theorems==
Every [[Compact space#Metric spaces|compact metric space]] is complete, though complete spaces need not be compact. In fact, a metric space is compact [[if and only if]] it is complete and [[totally bounded]]. This is a generalization of the [[Heine–Borel theorem]], which states that any closed and bounded subspace <math>S</math> of {{math|'''R'''<sup>''n''</sup>}} is compact and therefore complete.<ref>{{cite book |title=Introduction to Metric and Topological Spaces |first=Wilson A. |last=Sutherland|year=1975 |publisher=Clarendon Press |author-link= Wilson Sutherland |isbn=978-0-19-853161-6 }}</ref>

Let <math>(X, d)</math> be a complete metric space. If <math>A \subseteq X</math> is a closed set, then <math>A</math> is also complete. Let <math>(X, d)</math> be a metric space. If <math>A \subseteq X</math> is a complete subspace, then <math>A</math> is also closed.

If <math>X</math> is a [[set (mathematics)|set]] and <math>M</math> is a complete metric space, then the set <math>B(X, M)</math> of all bounded functions {{mvar|f}} from {{mvar|X}} to <math>M</math> is a complete metric space. Here we define the distance in <math>B(X, M)</math> in terms of the distance in <math>M</math> with the [[supremum norm]]
<math display=block>d(f, g) \equiv \sup\{d[f(x), g(x)]: x \in X\}</math>

If <math>X</math> is a [[topological space]] and <math>M</math> is a complete metric space, then the set <math>C_b(X, M)</math> consisting of all [[Continuous function (topology)|continuous]] bounded functions <math>f : X \to M</math> is a closed subspace of <math>B(X, M)</math> 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 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 (mathematics)|fixed point]]. The fixed-point theorem is often used to [[mathematical proof|prove]] the [[inverse function theorem]] on complete metric spaces such as Banach spaces.

{{Math theorem|name=Theorem<ref name="Zalinescu 2002 p. 33">{{cite book|last=Zalinescu|first=C.|title=Convex analysis in general vector spaces|publisher=World Scientific|publication-place=River Edge, N.J. London|year=2002|isbn=981-238-067-1|oclc=285163112|page=33}}</ref>|note=C. Ursescu|math_statement=
Let <math>X</math> be a complete metric space and let <math>S_1, S_2, \ldots</math> be a sequence of subsets of <math>X.</math>

* If each <math>S_i</math> is closed in <math>X</math> then <math display=inline>\operatorname{cl} \left(\bigcup_{i \in \N} \operatorname{int} S_i\right) = \operatorname{cl} \operatorname{int} \left(\bigcup_{i \in \N} S_i\right).</math>
* If each <math>S_i</math> is [[open subset|open]] in <math>X</math> then <math display=inline>\operatorname{int} \left(\bigcap_{i \in \N} \operatorname{cl} S_i\right) = \operatorname{int} \operatorname{cl} \left(\bigcap_{i \in \N} S_i\right).</math>
}}

==Completion==
For any metric space ''M'', it is possible to construct a complete metric space ''M′'' (which is also denoted as <math>\overline{M}</math>), 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 class]]es of Cauchy sequences in ''M''. For any two Cauchy sequences <math>x_{\bull} = \left(x_n\right)</math> and <math>y_{\bull} = \left(y_n\right)</math> in ''M'', we may define their distance as
<math display=block>d\left(x_{\bull}, y_{\bull}\right) = \lim_n d\left(x_n, y_n\right)</math>

(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 [[field extension|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 <math>p,</math> the [[p-adic number|{{math|''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==
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 {{open-open|0,1}}, which is not complete.

In [[topology]] one considers ''[[completely metrizable space]]s'', 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 space]]s.<ref>Kelley, Problem 6.L, p. 208</ref>

A topological space homeomorphic to a [[Separable space|separable]] complete metric space is called a [[Polish space]].

==Alternatives and generalizations==
{{Main|Uniform space#Completeness}}

Since [[Cauchy sequence]]s can also be defined in general [[topological group]]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]]s, but requires only the existence of a continuous "subtraction" operation.  In this setting, the distance between two points <math>x</math> and <math>y</math> is gauged not by a real number <math>\varepsilon</math> via the metric <math>d</math> in the comparison <math>d(x, y) < \varepsilon,</math> but by an [[open neighbourhood]] <math>N</math> of <math>0</math> via subtraction in the comparison <math>x - y \in N.</math>

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 Cauchy [[Filter (set theory)#Filter|filters]].  If every Cauchy net (or equivalently every Cauchy filter) has a limit in <math>X,</math> then <math>X</math> 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]]s; these too have a notion of completeness and completion just like uniform spaces.

==See also==

* {{annotated link|Cauchy space}}
* {{annotated link|Completion (algebra)}}
* {{annotated link|Complete uniform space}}
* {{annotated link|Complete topological vector space}}
* {{annotated link|Ekeland's variational principle}}
* {{annotated link|Knaster–Tarski theorem}}

==Notes==

{{reflist|group=note}}
{{reflist}}

==References==

* {{Cite book | last=Kelley | first=John L. |author-link=John L. Kelley| title=General Topology | isbn=0-387-90125-6 | publisher=Springer | year=1975}}
* [[Erwin Kreyszig|Kreyszig, Erwin]], ''Introductory functional analysis with applications'' (Wiley, New York, 1978). {{ISBN|0-471-03729-X}}
* [[Serge Lang|Lang, Serge]], "Real and Functional Analysis" {{ISBN|0-387-94001-4}}
* {{cite book
  | last1      = Meise
  | first1     = Reinhold
  | last2     =Vogt 
  | first2    = Dietmar
  | others=Ramanujan, M.S. (trans.)
  | title     = Introduction to functional analysis
  | publisher = Oxford: Clarendon Press; New York: Oxford University Press
  | year      = 1997
  | isbn      = 0-19-851485-9
}}

{{Metric spaces}}
{{DEFAULTSORT:Complete Metric Space}}

[[Category:Metric geometry]]
[[Category:Topology]]
[[Category:Uniform spaces]]