Editing Metric space (section)

==Basic notions==
A distance function is enough to define notions of closeness and convergence that were first developed in [[real analysis]].  Properties that depend on the structure of a metric space are referred to as ''metric properties''.  Every metric space is also a [[topological space]], and some metric properties can also be rephrased without reference to distance in the language of topology; that is, they are really [[topological property|topological properties]].

===The topology of a metric space===
For any point {{mvar|x}} in a metric space {{mvar|M}} and any real number {{math|''r'' > 0}}, the [[ball (mathematics)|''open ball'']] of radius {{mvar|r}} around {{mvar|x}} is defined to be the set of points that are strictly less than distance {{mvar|r}} from {{mvar|x}}:
<math display="block">B_r(x)=\{y \in M : d(x,y) < r\}.</math>
This is a natural way to define a set of points that are relatively close to {{mvar|x}}.  Therefore, a set <math>N \subseteq M</math> is a [[neighborhood (mathematics)|''neighborhood'']] of {{mvar|x}} (informally, it contains all points "close enough" to {{mvar|x}}) if it contains an open ball of radius {{mvar|r}} around {{mvar|x}} for some {{math|''r'' > 0}}.

An ''open set'' is a set which is a neighborhood of all its points.  It follows that the open balls form a [[base (topology)|base]] for a topology on {{mvar|M}}.  In other words, the open sets of {{mvar|M}} are exactly the unions of open balls.  As in any topology, [[closed set]]s are the complements of open sets.  Sets may be both open and closed as well as neither open nor closed.

This topology does not carry all the information about the metric space.  For example, the distances {{math|''d''<sub>1</sub>}}, {{math|''d''<sub>2</sub>}}, and {{math|''d''<sub>∞</sub>}} defined above all induce the same topology on <math>\R^2</math>, although they behave differently in many respects.  Similarly, <math>\R</math> with the Euclidean metric and its subspace the interval {{open-open|0, 1}} with the induced metric are [[homeomorphism|homeomorphic]] but have very different metric properties.

Conversely, not every topological space can be given a metric.  Topological spaces which are compatible with a metric are called [[metrizable space|''metrizable'']] and are particularly well-behaved in many ways: in particular, they are [[paracompact space|paracompact]]<ref>Rudin, Mary Ellen.  [https://www.jstor.org/stable/2035708 A new proof that metric spaces are paracompact] {{webarchive|url=https://web.archive.org/web/20160412015215/http://www.jstor.org/stable/2035708 |date=2016-04-12 }}.  Proceedings of the American Mathematical Society, Vol. 20, No. 2. (Feb., 1969), p. 603.</ref> [[Hausdorff space]]s (hence [[normal space|normal]]) and [[first-countable space|first-countable]].{{efn|Balls with rational radius around a point {{mvar|x}} form a [[neighborhood basis]] for that point.}}  The [[Nagata–Smirnov metrization theorem]] gives a characterization of metrizability in terms of other topological properties, without reference to metrics.

===Convergence===
[[limit of a sequence|Convergence of sequences]] in Euclidean space is defined as follows:
: A sequence {{math|(''x<sub>n</sub>'')}} converges to a point {{mvar|x}} if for every {{math|ε > 0}} there is an integer {{mvar|N}} such that for all {{math|''n'' > ''N''}}, {{math|''d''(''x<sub>n</sub>'', ''x'') < ε}}.
Convergence of sequences in a topological space is defined as follows:
: A sequence {{math|(''x<sub>n</sub>'')}} converges to a point {{mvar|x}} if for every open set {{mvar|U}} containing {{mvar|x}} there is an integer {{mvar|N}} such that for all {{math|''n'' > ''N''}}, <math>x_n \in U</math>.
In metric spaces, both of these definitions make sense and they are equivalent.  This is a general pattern for [[topological property|topological properties]] of metric spaces: while they can be defined in a purely topological way, there is often a way that uses the metric which is easier to state or more familiar from real analysis.

===Completeness===
{{main|Complete metric space}}
Informally, a metric space is ''complete'' if it has no "missing points": every sequence that looks like it should converge to something actually converges.

To make this precise: a sequence {{math|(''x<sub>n</sub>'')}} in a metric space {{mvar|M}} is [[Cauchy sequence|''Cauchy'']] if for every {{math|ε > 0}} there is an integer {{mvar|N}} such that for all {{math|''m'', ''n'' > ''N''}}, {{math|''d''(''x<sub>m</sub>'', ''x<sub>n</sub>'') < ε}}.  By the triangle inequality, any convergent sequence is Cauchy: if {{mvar|x<sub>m</sub>}} and {{mvar|x<sub>n</sub>}} are both less than {{math|ε}} away from the limit, then they are less than {{math|2ε}} away from each other.  If the converse is true—every Cauchy sequence in {{mvar|M}} converges—then {{mvar|M}} is complete.

Euclidean spaces are complete, as is <math>\R^2</math> with the other metrics described above.  Two examples of spaces which are not complete are {{open-open|0, 1}} and the rationals, each with the metric induced from <math>\R</math>.  One can think of {{open-open|0, 1}} as "missing" its endpoints 0 and 1.  The rationals are missing all the irrationals, since any irrational has a sequence of rationals converging to it in <math>\R</math> (for example, its successive decimal approximations).  These examples show that completeness is ''not'' a topological property, since <math>\R</math> is complete but the homeomorphic space {{open-open|0, 1}} is not.

This notion of "missing points" can be made precise.  In fact, every metric space has a unique [[completion (metric space)|''completion'']], which is a complete space that contains the given space as a [[dense set|dense]] subset. For example, {{closed-closed|0, 1}} is the completion of {{open-open|0, 1}}, and the real numbers are the completion of the rationals.

Since complete spaces are generally easier to work with, completions are important throughout mathematics.  For example, in abstract algebra, the [[p-adic numbers|''p''-adic numbers]] are defined as the completion of the rationals under a different metric.  Completion is particularly common as a tool in [[functional analysis]].  Often one has a set of nice functions and a way of measuring distances between them.  Taking the completion of this metric space gives a new set of functions which may be less nice, but nevertheless useful because they behave similarly to the original nice functions in important ways.  For example, [[weak solution]]s to [[differential equation]]s typically live in a completion (a [[Sobolev space]]) rather than the original space of nice functions for which the differential equation actually makes sense.
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If <math>X</math> is a complete subset of the metric space <math>M</math>, then <math>X</math> is closed in <math>M</math>. Indeed, a space is complete if and only if it is closed in any containing metric space.

Every complete metric space is a [[Baire space]].
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===Bounded and totally bounded spaces===
[[File:Diameter of a Set.svg|thumb|Diameter of a set.]]
{{See also|Bounded set|Diameter of a metric space}}
A metric space {{mvar|M}} is ''bounded'' if there is an {{mvar|r}} such that no pair of points in {{mvar|M}} is more than distance {{mvar|r}} apart.{{efn|In the context of [[Interval (mathematics)|interval]]s in the real line, or more generally regions in Euclidean space, bounded sets are sometimes referred to as "finite intervals" or "finite regions".  However, they do not typically have a finite number of elements, and while they all have finite [[Lebesgue measure|volume]], so do many unbounded sets.  Therefore this terminology is imprecise.}}  The least such {{mvar|r}} is called the [[diameter of a set|diameter]] of {{mvar|M}}.

The space {{mvar|M}} is called ''precompact'' or ''[[totally bounded]]'' if for every {{math|''r'' > 0}} there is a finite [[cover (topology)|cover]] of {{mvar|M}} by open balls of radius {{mvar|r}}.  Every totally bounded space is bounded.  To see this, start with a finite cover by {{mvar|r}}-balls for some arbitrary {{mvar|r}}.  Since the subset of {{mvar|M}} consisting of the centers of these balls is finite, it has finite diameter, say {{mvar|D}}.  By the triangle inequality, the diameter of the whole space is at most {{math|''D'' + 2''r''}}.  The converse does not hold: an example of a metric space that is bounded but not totally bounded is <math>\R^2</math> (or any other infinite set) with the discrete metric.

===Compactness===
{{Main|Compact space}}
Compactness is a topological property which generalizes the properties of a closed and bounded subset of Euclidean space.  There are several equivalent definitions of compactness in metric spaces:
# A metric space {{mvar|M}} is compact if every open cover has a finite subcover (the usual topological definition).
# A metric space {{mvar|M}} is compact if every sequence has a convergent subsequence.  (For general topological spaces this is called [[sequentially compact space|sequential compactness]] and is not equivalent to compactness.)
# A metric space {{mvar|M}} is compact if it is complete and totally bounded.  (This definition is written in terms of metric properties and does not make sense for a general topological space, but it is nevertheless topologically invariant since it is equivalent to compactness.)
One example of a compact space is the closed interval {{closed-closed|0, 1}}.

Compactness is important for similar reasons to completeness: it makes it easy to find limits.  Another important tool is [[Lebesgue's number lemma]], which shows that for any open cover of a compact space, every point is relatively deep inside one of the sets of the cover.
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Every compact metric space is [[second countable]], and is a continuous image of the [[Cantor set]]. (The latter result is due to [[Pavel Alexandrov]] and [[Pavel Samuilovich Urysohn|Urysohn]].)

===Locally compact and proper spaces===
A space is said to be ''[[locally compact]]'' if every point has a compact neighborhood. Euclidean spaces are locally compact, but infinite-dimensional [[Banach space]]s are not.

A metric space is [[proper space|''proper'']] if every ''closed ball'' <math>\{y\, \colon d(x,y)\leq r\}</math> is compact. A space is proper if and only if it is complete and locally compact.-->