TheInfoList

In
mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ...
, a metric space is a non empty set together with a
metric METRIC (Mapping EvapoTranspiration at high Resolution with Internalized Calibration) is a computer model Computer simulation is the process of mathematical modelling, performed on a computer, which is designed to predict the behaviour of or th ...

on the set. The metric is a
function Function or functionality may refer to: Computing * Function key A function key is a key on a computer A computer is a machine that can be programmed to carry out sequences of arithmetic or logical operations automatically. Modern comp ...
that defines a concept of ''distance'' between any two
members Member may refer to: * Military jury, referred to as "Members" in military jargon * Element (mathematics), an object that belongs to a mathematical set * In object-oriented programming, a member of a class ** Field (computer science), entries in ...
of the set, which are usually called
points Point or points may refer to: Places * Point, Lewis, a peninsula in the Outer Hebrides, Scotland * Point, Texas, a city in Rains County, Texas, United States * Point, the NE tip and a ferry terminal of Lismore, Scotland, Lismore, Inner Hebrides, ...
. The metric satisfies a few simple properties: * the distance from $A$ to $B$ is zero if and only if $A$ and $B$ are the same point, * the distance between two distinct points is positive, * the distance from $A$ to $B$ is the same as the distance from $B$ to $A$, and * the distance from $A$ to $B$ is less than or equal to the distance from $A$ to $B$ via any third point $C$. A metric on a space induces
topological propertiesIn topology s, which have only one surface and one edge, are a kind of object studied in topology. In mathematics, topology (from the Greek language, Greek words , and ) is concerned with the properties of a mathematical object, geometric object ...
like
open Open or OPEN may refer to: Music * Open (band) Open is a band. Background Drummer Pete Neville has been involved in the Sydney/Australian music scene for a number of years. He has recently completed a Masters in screen music at the Australia ...
and
closed set In geometry Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position of ...
s, which lead to the study of more abstract
topological 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 gener ...
s. The most familiar metric space is
3-dimensional Euclidean space Three-dimensional space (also: 3-space or, rarely, tri-dimensional space) is a geometric setting in which three values (called parameter A parameter (from the Ancient Greek language, Ancient Greek wikt:παρά#Ancient Greek, παρά, ''par ...
. In fact, a "metric" is the generalization of the
Euclidean metric In mathematics, the Euclidean distance between two points in Euclidean space is the length of a line segment between the two points. It can be calculated from the Cartesian coordinates of the points using the Pythagorean theorem, therefore occas ...
arising from the four long-known properties of the Euclidean distance. The Euclidean metric defines the distance between two points as the length of the
line segment In geometry Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position ...

connecting them. Other metric spaces occur for example in
elliptic geometry Elliptic geometry is an example of a geometry Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is conc ...
and
hyperbolic geometry 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 g ...
, where distance on a
sphere A sphere (from Greek#REDIRECT Greek Greek may refer to: Greece Anything of, from, or related to Greece Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a country located in Southeast Europe. Its population is appr ...

measured by angle is a metric, and the
hyperboloid model In geometry Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space th ...
of hyperbolic geometry is used by
special relativity In physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion and behavior through Spacetime, space and time, and the related entities of energy and force ...
as a metric space of
velocities The velocity of an object is the Time derivative, rate of change of its Position (vector), position with respect to a frame of reference, and is a function of time. Velocity is equivalent to a specification of an object's speed and direction o ...

. Some of non-geometric metric spaces include spaces of finite strings (
finite 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 ...
s of symbols from a predefined alphabet) equipped with e.g. a
Hamming Hamming may refer to: * Richard Hamming Richard Wesley Hamming (February 11, 1915 – January 7, 1998) was an American mathematician whose work had many implications for computer engineering Computer engineering (CoE or CpE) is a branch o ...
's or
Levenshtein distance In information theory Information theory is the scientific study of the quantification, storage, and communication Communication (from Latin ''communicare'', meaning "to share") is the act of developing Semantics, meaning among Subject (phi ...
, a space of subsets of any metric space equipped with
Hausdorff distanceIn mathematics, the Hausdorff distance, or Hausdorff metric, also called Pompeiu–Hausdorff distance, measures how far two subsets of a metric space are from each other. It turns the set of non-empty set, non-empty compact space, compact subsets of ...
, a space of real functions
integrableIn mathematics, integrability is a property of certain dynamical system In mathematics, a dynamical system is a system in which a Function (mathematics), function describes the time dependence of a Point (geometry), point in a Manifold, geometric ...

on a unit interval with an integral metric $d\left(f,g\right)=\int_^\left\vert f\left(x\right)-g\left(x\right)\right\vert\,dx$ or probabilistic spaces on any chosen metric space equipped with
Wasserstein metric 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 ...

# History

In 1906
Maurice FréchetMaurice may refer to: People *Saint Maurice Saint Maurice (also Moritz, Morris, or Mauritius; ) was the leader of the legendary Roman Theban Legion in the 3rd century, and one of the favorite and most widely venerated saints of that group. He w ...
introduced metric spaces in his work ''Sur quelques points du calcul fonctionnel''. However the name is due to
Felix Hausdorff Felix Hausdorff (November 8, 1868 – January 26, 1942) was a German mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as q ...
.

# Definition

A metric space is an
ordered pair 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 ...

$\left(M,d\right)$ where $M$ is a set and $d$ is a
metric METRIC (Mapping EvapoTranspiration at high Resolution with Internalized Calibration) is a computer model Computer simulation is the process of mathematical modelling, performed on a computer, which is designed to predict the behaviour of or th ...
on $M$, i.e., a
function Function or functionality may refer to: Computing * Function key A function key is a key on a computer A computer is a machine that can be programmed to carry out sequences of arithmetic or logical operations automatically. Modern comp ...
:$d\,\colon M \times M \to \mathbb$ such that for any $x, y, z \in M$, the following holds: : Given the above three axioms, we also have that $d\left(x,y\right) \ge 0$ for any $x, y \in M$. This is deduced as follows (from the top to the bottom): : The function $d$ is also called ''distance function'' or simply ''distance''. Often, $d$ is omitted and one just writes $M$ for a metric space if it is clear from the context what metric is used. Ignoring mathematical details, for any system of roads and terrains the distance between two locations can be defined as the length of the shortest route connecting those locations. To be a metric there shouldn't be any one-way roads. The triangle inequality expresses the fact that detours aren't shortcuts. If the distance between two points is zero, the two points are indistinguishable from one-another. Many of the examples below can be seen as concrete versions of this general idea.

# Examples of metric spaces

* The
real 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 g ...
s with the distance function $d\left(x,y\right) = , y - x ,$ given by the
absolute difference The absolute difference of two 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 ...

, and, more generally, Euclidean -space with the
Euclidean distance 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). ...
, are complete metric spaces. The
rational number In mathematics, a rational number is a number that can be expressed as the quotient or fraction (mathematics), fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ) ...
s with the same distance function also form a metric space, but not a complete one. * The
positive real numbersIn 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 ha ...
with distance function $d\left(x,y\right) =\left, \log\left(y/x\right) \$ is a complete metric space. * Any
normed vector 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 g ...
is a metric space by defining $d\left(x,y\right) = \lVert y - x \rVert$, see also metrics on vector spaces. (If such a space is complete, we call it 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 ...
.) Examples: ** The Manhattan norm gives rise to the
Manhattan distance A taxicab geometry is a form of geometry in which the usual distance function or Metric (mathematics), metric of Euclidean geometry is replaced by a new metric in which the distance between two points is the sum of the absolute differences of thei ...

, where the distance between any two points, or vectors, is the sum of the differences between corresponding coordinates. ** The cyclic Mannheim metric or Mannheim distance is a modulo variant of the Manhattan metric. ** The
maximum 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 functions ''f'' defined on ...
gives rise to the
Chebyshev distance In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
or chessboard distance, the minimal number of moves a chess king would take to travel from $x$ to $y$. * The
British Rail British Railways (BR), which from 1965 traded as British Rail, was the state-owned company that operated the national rail system rail transport in Great Britain The railway system in Great Britain is the oldest railway system in the world. ...
metric (also called the "post office metric" or the "
SNCF The Société nationale des chemins de fer français (; abbreviated as SNCF ; French for "French National Railway Company") is France France (), officially the French Republic (french: link=no, République française), is a List of tr ...

metric") on a
normed vector 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 g ...
is given by $d\left(x,y\right) = \lVert x \rVert + \lVert y \rVert$ for distinct points $x$ and $y$, and $d\left(x,x\right) = 0$. More generally $\lVert \cdot \rVert$ can be replaced with a function $f$ taking an arbitrary set $S$ to non-negative reals and taking the value $0$ at most once: then the metric is defined on $S$ by $d\left(x,y\right) = f\left(x\right) + f\left(y\right)$ for distinct points $x$ and $y$, and The name alludes to the tendency of railway journeys to proceed via London (or Paris) irrespective of their final destination. * If $\left(M,d\right)$ is a metric space and $X$ is a
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). ...

of then $\left(X,d\right)$ becomes a metric space by restricting the domain of $d$ to * The
discrete metric Discrete in science is the opposite of continuous Continuity or continuous may refer to: Mathematics * Continuity (mathematics), the opposing concept to discreteness; common examples include ** Continuous probability distribution or random vari ...
, where $d\left(x,y\right) = 0$ if $x=y$ and $d\left(x,y\right) = 1$ otherwise, is a simple but important example, and can be applied to all sets. This, in particular, shows that for any set, there is always a metric space associated to it. Using this metric, the singleton of any point is an
open ball 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 ...
, therefore every subset is open and the space has the
discrete topology In topology s, which have only one surface and one edge, are a kind of object studied in topology. In mathematics, topology (from the Greek language, Greek words , and ) is concerned with the properties of a mathematical object, geometric objec ...
. * A finite metric space is a metric space having a
finite Finite is the opposite of Infinity, infinite. It may refer to: * Finite number (disambiguation) * Finite set, a set whose cardinality (number of elements) is some natural number * Finite verb, a verb form that has a subject, usually being inflected ...
number of points. Not every finite metric space can be isometrically embedded in a
Euclidean space Euclidean space is the fundamental space of classical geometry. Originally, it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean spaces of any nonnegative integer dimension (mathematics), dimens ...
. * The
hyperbolic plane 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). ...
is a metric space. More generally: ** If $M$ is any connected
Riemannian manifold In differential geometry Differential geometry is a mathematical Mathematics (from Greek Greek may refer to: Greece Anything of, from, or related to Greece Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a c ...
, then we can turn $M$ into a metric space by defining the distance of two points as the
infimum In mathematics, the infimum (abbreviated inf; plural infima) of a subset S of a partially ordered set P is a greatest element in P that is less than or equal to all elements of S, if such an element exists. Consequently, the term ''greatest low ...
of the lengths of the paths (continuously differentiable
curve In mathematics, a curve (also called a curved line in older texts) is an object similar to a line (geometry), line, but that does not have to be Linearity, straight. Intuitively, a curve may be thought of as the trace left by a moving point (geo ...

s) connecting them. * If $X$ is some set and $M$ is a metric space, then, the set of all
bounded function In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
s $f \colon X \to M$ (i.e. those functions whose image is a bounded subset of $M$) can be turned into a metric space by defining $d\left(f,g\right) = \sup_ d\left(f\left(x\right),g\left(x\right)\right)$ for any two bounded functions $f$ and $g$ (where $\sup$ is
supremum 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 ...

). This metric is called the or supremum metric, and If $M$ is complete, then this
function space In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and th ...
is complete as well. If ''X'' is also a topological space, then the set of all bounded
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 ga ...
functions from $X$ to $M$ (endowed with the uniform metric), will also be a complete metric if ''M'' is. * If $G$ is an undirected connected graph, then the set $V$ of vertices of $G$ can be turned into a metric space by defining $d\left(x,y\right)$ to be the length of the shortest path connecting the vertices $x$ and In
geometric group theory Geometric group theory is an area 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 (mat ...
this is applied to the
Cayley graph on two generators ''a'' and ''b'' In mathematics, a Cayley graph, also known as a Cayley colour graph, Cayley diagram, group diagram, or colour group is a Graph (discrete mathematics), graph that encodes the abstract structure of a group (mathemati ...

of a group, yielding the
word metricIn group theory The popular puzzle Rubik's cube invented in 1974 by Ernő Rubik has been used as an illustration of permutation group">Ernő_Rubik.html" ;"title="Rubik's cube invented in 1974 by Ernő Rubik">Rubik's cube invented in 1974 by Ern ...
. *
Graph edit distanceIn mathematics and computer science, graph edit distance (GED) is a Similarity measure, measure of similarity (or dissimilarity) between two Graph (discrete mathematics), graphs. The concept of graph edit distance was first formalized mathematically ...
is a measure of dissimilarity between two
graphs Graph may refer to: Mathematics *Graph (discrete mathematics) In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes an ...
, defined as the minimal number of graph edit operations required to transform one graph into another. * The
Levenshtein distance In information theory Information theory is the scientific study of the quantification, storage, and communication Communication (from Latin ''communicare'', meaning "to share") is the act of developing Semantics, meaning among Subject (phi ...
is a measure of the dissimilarity between two
strings String or strings may refer to: *String (structure), a long flexible structure made from threads twisted together, which is used to tie, bind, or hang other objects Arts, entertainment, and media Films * Strings (1991 film), ''Strings'' (1991 fil ...
$u$ and $v$, defined as the minimal number of character deletions, insertions, or substitutions required to transform $u$ into $v$. This can be thought of as a special case of the shortest path metric in a graph and is one example of an
edit distance In computational linguistics and computer science, edit distance is a way of quantifying how dissimilar two String (computing), strings (e.g., words) are to one another by counting the minimum number of operations required to transform one string in ...
. * Given a metric space $\left(X,d\right)$ and an increasing
concave function In , a concave function is the of a . A concave function is also ously called concave downwards, concave down, convex upwards, convex cap, or upper convex. Definition A real-valued f on an (or, more generally, a in ) is said to be ''concave' ...

such that $f\left(x\right) = 0$ if and only if $x=0$, then $f \circ d$ is also a metric on $X$. * Given an injective function $f$ from any set $A$ to a metric space $\left(X,d\right)$, $d\left(f\left(x\right), f\left(y\right)\right)$ defines a metric on $A$. * Using T-theory, the tight span of a metric space is also a metric space. The tight span is useful in several types of analysis. * The set of all $m$ by $n$ matrix (mathematics), matrices over some field (mathematics), field is a metric space with respect to the
rank Rank is the relative position, value, worth, complexity, power, importance, authority, level, etc. of a person or object within a ranking A ranking is a relationship between a set of items such that, for any two items, the first is either "rank ...
distance $d\left(X,Y\right) = \mathrm\left(Y - X\right)$. * The
Helly metric In game theory Game theory is the study of mathematical model A mathematical model is a description of a system A system is a group of Interaction, interacting or interrelated elements that act according to a set of rules to form a u ...
is used in
game theory Game theory is the study of mathematical model A mathematical model is a description of a system A system is a group of Interaction, interacting or interrelated elements that act according to a set of rules to form a unified whole. ...
.

# Open and closed sets, topology and convergence

Every metric space is a
topological 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 gener ...
in a natural manner, and therefore all definitions and theorems about general topological spaces also apply to all metric spaces. About any point $x$ in a metric space $M$ we define the
open ball 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 radius $r > 0$ (where $r$ is a real number) about $x$ as the set :$B\left(x;r\right) = \.$ These open balls form the
base Base or BASE may refer to: Brands and enterprises * Base (mobile telephony provider), a Belgian mobile telecommunications operator *Base CRM Base CRM (originally Future Simple or PipeJump) is an enterprise software company based in Mountain Vie ...
for a topology on ''M'', making it a
topological 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 gener ...
. Explicitly, a subset $U$ of $M$ is called
open Open or OPEN may refer to: Music * Open (band) Open is a band. Background Drummer Pete Neville has been involved in the Sydney/Australian music scene for a number of years. He has recently completed a Masters in screen music at the Australia ...
if for every $x$ in $U$ there exists an $r > 0$ such that $B\left(x;r\right)$ is contained in $U$. The
complement A complement is often something that completes something else, or at least adds to it in some useful way. Thus it may be: * Complement (linguistics), a word or phrase having a particular syntactic role ** Subject complement, a word or phrase addi ...
of an open set is called closed. A
neighborhood A neighbourhood (British English British English (BrE) is the standard dialect A standard language (also standard variety, standard dialect, and standard) is a language variety that has undergone substantial codification of grammar ...
of the point $x$ is any subset of $M$ that contains an open ball about $x$ as a subset. A topological space which can arise in this way from a metric space is called a
metrizable space In topology and related areas of mathematics, a metrizable space is a topological space that is Homeomorphism, homeomorphic to a metric space. That is, a topological space (X, \mathcal) is said to be metrizable if there is a Metric (mathematics), m ...
. A
sequence In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...

($x_n$) in a metric space $M$ is said to
converge Converge may refer to: * Converge (band), American hardcore punk band * Converge (Baptist denomination), American national evangelical Baptist body * Limit (mathematics) * Converge ICT, internet service provider in the Philippines See also

...
to the limit $x \in M$
if and only if In logic Logic is an interdisciplinary field which studies truth and reasoning. Informal logic seeks to characterize Validity (logic), valid arguments informally, for instance by listing varieties of fallacies. Formal logic represents st ...
for every $\varepsilon>0$, there exists a natural number ''N'' such that $d\left(x_n,x\right) < \varepsilon$ for all $n > N$. Equivalently, one can use the general definition of convergence available in all topological spaces. A subset $A$ of the metric space $M$ is closed if and only if every sequence in $A$ that converges to a limit in $M$ has its limit in $A$.

# Types of metric spaces

## Complete spaces

A metric space $M$ is said to be complete if every
Cauchy sequence In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...
converges in $M$. That is to say: if $d\left(x_n, x_m\right) \to 0$ as both $n$ and $m$ independently go to infinity, then there is some $y\in M$ with $d\left(x_n, y\right) \to 0$. Every
Euclidean space Euclidean space is the fundamental space of classical geometry. Originally, it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean spaces of any nonnegative integer dimension (mathematics), dimens ...
is complete, as is every closed subset of a complete space. The rational numbers, using the absolute value metric $d\left(x,y\right) = \vert x - y \vert$, are not complete. Every metric space has a unique (up to
isometry In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...
) completion, which is a complete space that contains the given space as a
dense The density (more precisely, the volumetric mass density; also known as specific mass), of a substance is its mass Mass is both a property Property (''latin: Res Privata'') in the Abstract and concrete, abstract is what belongs to or ...
subset. For example, the real numbers are the completion of the rationals. If $X$ is a complete subset of the metric space $M$, then $X$ is closed in $M$. 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 In mathematics, a Baire space is a topological 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 ...
.

## Bounded and totally bounded spaces

A metric space $M$ is called if there exists some number $r$, such that $d\left(x, y\right) \leq r$ for all $x, y \in M.$ The smallest possible such $r$ is called the of $M.$ The space $M$ is called precompact or
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 ...
if for every $r>0$ there exist finitely many open balls of radius $r$ whose union covers $M.$ Since the set of the centres of these balls is finite, it has finite diameter, from which it follows (using the triangle inequality) that every totally bounded space is bounded. The converse does not hold, since any infinite set can be given the discrete metric (one of the examples above) under which it is bounded and yet not totally bounded. Note that in the context of intervals in the space of
real 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 g ...
s and occasionally regions in a Euclidean space $\R^n$ a bounded set is referred to as "a finite interval" or "finite region". However boundedness should not in general be confused with "finite", which refers to the number of elements, not to how far the set extends; finiteness implies boundedness, but not conversely. Also note that an unbounded subset of $\R^n$ may have a finite
volume Volume is a scalar quantity expressing the amount Quantity or amount is a property that can exist as a multitude Multitude is a term for a group of people who cannot be classed under any other distinct category, except for their shared fact ...
.

## Compact spaces

A metric space $M$ is compact if every sequence in $M$ has a
subsequence 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 ...
that converges to a point in $M$. This is known as sequential compactness and, in metric spaces (but not in general topological spaces), is equivalent to the topological notions of countable compactness and
compactness 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 ...
defined via
open cover 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. Examples of compact metric spaces include the closed interval
Cantor set In mathematics, the Cantor set is a set of points lying on a single line segment that has a number of remarkable and deep properties. It was discovered in 1874 by Henry John Stephen Smith and introduced by German mathematician Georg Cantor in 188 ...

. Every closed subset of a compact space is itself compact. A metric space is compact if and only if it is complete and totally bounded. This is known as the
Heine–Borel theorem In real analysis the Heine–Borel theorem, named after Eduard Heine and Émile Borel, states: For a subset ''S'' of Euclidean space R''n'', the following two statements are equivalent: *''S'' is closed set, closed and bounded set, bounded *''S'' i ...
. Note that compactness depends only on the topology, while boundedness depends on the metric. Lebesgue's number lemma states that for every open cover of a compact metric space $M$, there exists a "Lebesgue number" $\delta$ such that every subset of $M$ of
diameter In geometry Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position ...

$r<\delta$ is contained in some member of the cover. Every compact metric space is
second countable In 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 struc ...
, and is a continuous image of the
Cantor set In mathematics, the Cantor set is a set of points lying on a single line segment that has a number of remarkable and deep properties. It was discovered in 1874 by Henry John Stephen Smith and introduced by German mathematician Georg Cantor in 188 ...

. (The latter result is due to
Pavel Alexandrov Pavel Sergeyevich Alexandrov (russian: Па́вел Серге́евич Алекса́ндров), sometimes romanized ''Paul Alexandroff'' (7 May 1896 – 16 November 1982), was a Soviet The Soviet Union,. officially the Union of Soviet ...
and
Urysohn Pavel Samuilovich Urysohn () (February 3, 1898 – August 17, 1924) was a Soviet Union, Soviet mathematician who is best known for his contributions in dimension theory, and for developing Urysohn's metrization theorem and Urysohn's lemma, both of ...
.)

## Locally compact and proper spaces

A metric space is said to be
locally compact In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and the ...
if every point has a compact neighborhood. Euclidean spaces are locally compact, but infinite-dimensional Banach spaces are not. A space is proper if every closed ball $\$ is compact. Proper spaces are locally compact, but the converse is not true in general.

## Connectedness

A metric space $M$ is connected if the only subsets that are both open and closed are the empty set and $M$ itself. A metric space $M$ is
path connected In topology and related branches of mathematics, a connected space is a topological space that cannot be represented as the union (set theory), union of two or more disjoint set, disjoint Empty set, non-empty open (topology), open subsets. Conne ...
if for any two points $x, y \in M$ there exists a continuous map with $f\left(0\right)=x$ and $f\left(1\right)=y$. Every path connected space is connected, but the converse is not true in general. There are also local versions of these definitions:
locally connected spaceImage:Neighborhood illust1.svg, In this topological space, ''V'' is a neighbourhood of ''p'' and it contains a connected open set (the dark green disk) that contains ''p''. In topology and other branches of mathematics, a topological space ''X'' is ...
s and locally path connected spaces.
Simply connected space In topology, a topological space is called simply connected (or 1-connected, or 1-simply connected) if it is path-connected and every Path (topology), path between two points can be continuously transformed (intuitively for embedded spaces, staying ...
s are those that, in a certain sense, do not have "holes".

## Separable spaces

A metric space is
separable 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 ...
if it has a
countable In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
dense The density (more precisely, the volumetric mass density; also known as specific mass), of a substance is its mass Mass is both a property Property (''latin: Res Privata'') in the Abstract and concrete, abstract is what belongs to or ...
subset. Typical examples are the real numbers or any Euclidean space. For metric spaces (but not for general topological spaces) separability is equivalent to
second-countability In topology, a second-countable space, also called a completely separable space, is a topological space whose topology has a countable base (topology), base. More explicitly, a topological space T is second-countable if there exists some countable ...
and also to the Lindelöf property.

## Pointed metric spaces

If $X$ is a metric space and $x_0\in X$ then $\left(X,x_0\right)$ is called a ''pointed metric space'', and $x_0$ is called a ''distinguished point''. Note that a pointed metric space is just a nonempty metric space with attention drawn to its distinguished point, and that any nonempty metric space can be viewed as a pointed metric space. The distinguished point is sometimes denoted $0$ due to its similar behavior to zero in certain contexts.

# Types of maps between metric spaces

Suppose $\left(M_1,d_1\right)$ and $\left(M_2,d_2\right)$ are two metric spaces.

## Continuous maps

The map $f\,\colon M_1\to M_2$ is
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 ga ...
if it has one (and therefore all) of the following equivalent properties: ;General topological continuity: for every open set $U$ in $M_2$, the
preimage 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). ...
Eduard Heine Heinrich Eduard Heine (16 March 1821 – October 1881) was a German German(s) may refer to: Common uses * of or related to Germany * Germans, Germanic ethnic group, citizens of Germany or people of German ancestry * For citizens of German ...
. ;ε-δ definition: for every $x\in M_1$ and every $\varepsilon>0$ there exists $\delta>0$ such that for all $y$ in $M_1$ we have ::$d_1\left(x,y\right)<\delta \implies d_2\left(f\left(x\right),f\left(y\right)\right)< \varepsilon.$ :This uses the (ε, δ)-definition of limit, and is due to Augustin Louis Cauchy. Moreover, $f$ is continuous if and only if it is continuous on every compact subset of $M_1$. The image (mathematics), image of every compact set under a continuous function is compact, and the image of every connected set under a continuous function is connected.

## Uniformly continuous maps

The map $f\,\colon M_1\to M_2$ is uniform continuity, uniformly continuous if for every $\varepsilon>0$ there exists $\delta>0$ such that :$d_1\left(x,y\right)<\delta \implies d_2\left(f\left(x\right),f\left(y\right)\right)< \varepsilon \quad\mbox\quad x,y\in M_1.$ Every uniformly continuous map $f\,\colon M_1\to M_2$ is continuous. The converse is true if $M_1$ is compact (Heine–Cantor theorem). Uniformly continuous maps turn Cauchy sequences in $M_1$ into Cauchy sequences in $M_2$. For continuous maps this is generally wrong; for example, a continuous map from the open interval $\left(0,1\right)$ ''onto'' the real line turns some Cauchy sequences into unbounded sequences.

## Lipschitz-continuous maps and contractions

Given a real number $K>0$, the map $f\,\colon M_1\to M_2$ is Lipschitz continuity, ''K''-Lipschitz continuous if :$d_2\left(f\left(x\right),f\left(y\right)\right)\leq K d_1\left(x,y\right)\quad\mbox\quad x,y\in M_1.$ Every Lipschitz-continuous map is uniformly continuous, but the converse is not true in general. If $K<1$, then $f$ is called a contraction mapping, contraction. Suppose $M_2=M_1$ and $M_1$ is complete. If $f$ is a contraction, then $f$ admits a unique fixed point (Banach fixed-point theorem). If $M_1$ is compact, the condition can be weakened a bit: $f$ admits a unique fixed point if :$d\left(f\left(x\right), f\left(y\right)\right) < d\left(x, y\right) \quad \mbox \quad x \ne y \in M_1$.

## Isometries

The map $f\,\colon M_1\to M_2$ is an
isometry In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...
if :$d_2\left(f\left(x\right),f\left(y\right)\right)=d_1\left(x,y\right)\quad\mbox\quad x,y\in M_1$ Isometries are always injective; the image of a compact or complete set under an isometry is compact or complete, respectively. However, if the isometry is not surjective, then the image of a closed (or open) set need not be closed (or open).

## Quasi-isometries

The map $f\,\colon M_1\to M_2$ is a quasi-isometry if there exist constants $A\geq1$ and $B\geq0$ such that :$\frac d_2\left(f\left(x\right),f\left(y\right)\right)-B\leq d_1\left(x,y\right)\leq A d_2\left(f\left(x\right),f\left(y\right)\right)+B \quad\text\quad x,y\in M_1$ and a constant $C\geq0$ such that every point in $M_2$ has a distance at most $C$ from some point in the image $f\left(M_1\right)$. Note that a quasi-isometry is not required to be continuous. Quasi-isometries compare the "large-scale structure" of metric spaces; they find use in
geometric group theory Geometric group theory is an area 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 (mat ...
in relation to the
word metricIn group theory The popular puzzle Rubik's cube invented in 1974 by Ernő Rubik has been used as an illustration of permutation group">Ernő_Rubik.html" ;"title="Rubik's cube invented in 1974 by Ernő Rubik">Rubik's cube invented in 1974 by Ern ...
.

# Notions of metric space equivalence

Given two metric spaces $\left(M_1, d_1\right)$ and $\left(M_2, d_2\right)$: *They are called homeomorphic (topologically isomorphic) if there exists a homeomorphism between them (i.e., a bijection continuous in both directions). *They are called uniformic (uniformly isomorphic) if there exists a uniform isomorphism between them (i.e., a bijection uniformly continuous in both directions). *They are called isometric if there exists a bijective
isometry In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...
between them. In this case, the two metric spaces are essentially identical. *They are called quasi-isometric if there exists a quasi-isometry between them.

# Topological properties

Metric spaces are paracompact Hausdorff spaces and hence normal space, normal (indeed they are perfectly normal). An important consequence is that every metric space admits partitions of unity and that every continuous real-valued function defined on a closed subset of a metric space can be extended to a continuous map on the whole space (Tietze extension theorem). It is also true that every real-valued Lipschitz maps, Lipschitz-continuous map defined on a subset of a metric space can be extended to a Lipschitz-continuous map on the whole space. Metric spaces are first countable since one can use balls with rational radius as a neighborhood base. The metric topology on a metric space $M$ is the coarsest topology on $M$ relative to which the metric $d$ is a continuous map from the product of $M$ with itself to the non-negative real numbers.

# Distance between points and sets; Hausdorff distance and Gromov metric

A simple way to construct a function separating a point from a closed set (as required for a completely regular space) is to consider the Distance#Distances between sets and between a point and a set, distance between the point and the set. If $\left(M,d\right)$ is a metric space, $S$ is a
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). ...

of $M$ and $x$ is a point of $M$, we define the distance from $x$ to $S$ as :$d\left(x,S\right) = \inf\$ where $\inf$ represents the
infimum In mathematics, the infimum (abbreviated inf; plural infima) of a subset S of a partially ordered set P is a greatest element in P that is less than or equal to all elements of S, if such an element exists. Consequently, the term ''greatest low ...
. Then $d\left(x, S\right)=0$ if and only if $x$ belongs to the closure (topology), closure of $S$. Furthermore, we have the following generalization of the triangle inequality: :$d\left(x,S\right) \leq d\left(x,y\right) + d\left(y,S\right),$ which in particular shows that the map $x\mapsto d\left(x,S\right)$ is continuous. Given two subsets $S$ and $T$ of $M$, we define their
Hausdorff distanceIn mathematics, the Hausdorff distance, or Hausdorff metric, also called Pompeiu–Hausdorff distance, measures how far two subsets of a metric space are from each other. It turns the set of non-empty set, non-empty compact space, compact subsets of ...
to be :$d_H\left(S,T\right) = \max \$ where $\sup$ represents the
supremum 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 ...

. In general, the Hausdorff distance $d_H\left(S,T\right)$ can be infinite. Two sets are close to each other in the Hausdorff distance if every element of either set is close to some element of the other set. The Hausdorff distance $d_H$ turns the set $K\left(M\right)$ of all non-empty compact subsets of $M$ into a metric space. One can show that $K\left(M\right)$ is complete if $M$ is complete. (A different notion of convergence of compact subsets is given by the Kuratowski convergence.) One can then define the Gromov–Hausdorff convergence, Gromov–Hausdorff distance between any two metric spaces by considering the minimal Hausdorff distance of isometrically embedded versions of the two spaces. Using this distance, the class of all (isometry classes of) compact metric spaces becomes a metric space in its own right.

# Product metric spaces

If $\left(M_1,d_1\right),\ldots,\left(M_n,d_n\right)$ are metric spaces, and $N$ is the Euclidean norm on $\mathbb R^n$, then $\Bigl\left(M_1 \times \cdots \times M_n, N\left(d_1,\ldots,d_n\right)\Bigr\right)$ is a metric space, where the product metric is defined by :$N\left(d_1,\ldots,d_n\right)\Big\left(\left(x_1,\ldots,x_n\right),\left(y_1,\ldots,y_n\right)\Big\right) = N\Big\left(d_1\left(x_1,y_1\right),\ldots,d_n\left(x_n,y_n\right)\Big\right),$ and the induced topology agrees with the product topology. By the equivalence of norms in finite dimensions, an equivalent metric is obtained if $N$ is the Norm (mathematics)#Taxicab norm or Manhattan norm, taxicab norm, a Norm (mathematics)#p-norm, p-norm, the maximum norm, or any other norm which is non-decreasing as the coordinates of a positive $n$-tuple increase (yielding the triangle inequality). Similarly, a countable product of metric spaces can be obtained using the following metric :$d\left(x,y\right)=\sum_^\infty \frac1\frac.$ An uncountable product of metric spaces need not be metrizable. For example, $\mathbb^\mathbb$ is not first-countable space, first-countable and thus isn't metrizable.

## Continuity of distance

In the case of a single space $\left(M,d\right)$, the distance map $d\colon M\times M \to R^+$ (from the Metric space#Definition, definition) is uniformly continuous with respect to any of the above product metrics $N\left(d,d\right)$, and in particular is continuous with respect to the product topology of $M\times M$.

# Quotient metric spaces

If ''M'' is a metric space with metric $d$, and $\sim$ is an equivalence relation on $M$, then we can endow the quotient set $M/\!\sim$ with a pseudometric. Given two equivalence classes $\left[x\right]$ and $\left[y\right]$, we define :$d\text{'}\left(\left[x\right],\left[y\right]\right) = \inf\$ where the
infimum In mathematics, the infimum (abbreviated inf; plural infima) of a subset S of a partially ordered set P is a greatest element in P that is less than or equal to all elements of S, if such an element exists. Consequently, the term ''greatest low ...
is taken over all finite sequences $\left(p_1, p_2, \dots, p_n\right)$ and $\left(q_1, q_2, \dots, q_n\right)$ with $\left[p_1\right]=\left[x\right]$, $\left[q_n\right]=\left[y\right]$, $\left[q_i\right]=\left[p_\right], i=1,2,\dots, n-1$. In general this will only define a pseudometric space, pseudometric, i.e. $d\text{'}\left(\left[x\right],\left[y\right]\right)=0$ does not necessarily imply that $\left[x\right]=\left[y\right]$. However, for some equivalence relations (e.g., those given by gluing together polyhedra along faces), $d\text{'}$ is a metric. The quotient metric $d$ is characterized by the following universal property. If $f\,\colon\left(M,d\right)\to\left(X,\delta\right)$ is a metric map between metric spaces (that is, $\delta\left(f\left(x\right),f\left(y\right)\right)\le d\left(x,y\right)$ for all $x$, $y$) satisfying $f\left(x\right)=f\left(y\right)$ whenever $x\sim y,$ then the induced function $\overline\,\colon M/\!\sim\to X$, given by $\overline\left(\left[x\right]\right)=f\left(x\right)$, is a metric map $\overline\,\colon \left(M/\!\sim,d\text{'}\right)\to \left(X,\delta\right).$ A topological space is sequential space, sequential if and only if it is a quotient of a metric space.

# Generalizations of metric spaces

* Every metric space is a uniform space in a natural manner, and every uniform space is naturally a
topological 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 gener ...
. Uniform and topological spaces can therefore be regarded as generalizations of metric spaces. * Relaxing the requirement that the distance between two distinct points be non-zero leads to the concepts of a pseudometric space or a dislocated metric space. Removing the requirement of symmetry, we arrive at a quasimetric space. Replacing the triangle inequality with a weaker form leads to semimetric spaces. * If the distance function takes values in the extended real number line $\mathbb R\cup\$, but otherwise satisfies the conditions of a metric, then it is called an ''extended metric'' and the corresponding space is called an ''$\infty$-metric space''. If the distance function takes values in some (suitable) ordered set (and the triangle inequality is adjusted accordingly), then we arrive at the notion of ''generalized ultrametric''. * Approach spaces are a generalization of metric spaces, based on point-to-set distances, instead of point-to-point distances. * A continuity space is a generalization of metric spaces and posets, that can be used to unify the notions of metric spaces and Domain theory, domains. * A partial metric space is intended to be the least generalisation of the notion of a metric space, such that the distance of each point from itself is no longer necessarily zero.

## Metric spaces as enriched categories

The ordered set $\left(\mathbb,\geq\right)$ can be seen as a Category (mathematics), category by requesting exactly one morphism $a\to b$ if $a\geq b$ and none otherwise. By using $+$ as the tensor product and $0$ as the Identity element, identity, it becomes a monoidal category $R^*$. Every metric space $\left(M,d\right)$ can now be viewed as a category $M^*$ enriched category, enriched over $R^*$: * Set $\operatorname\left(M^*\right):=M$ * For each $X,Y\in M$ set $\operatorname\left(X,Y\right):=d\left(X,Y\right)\in \operatorname\left(R^*\right)$ * The composition morphism $\operatorname\left(Y,Z\right)\otimes \operatorname\left(X,Y\right)\to \operatorname\left(X,Z\right)$ will be the unique morphism in $R^*$ given from the triangle inequality $d\left(y,z\right)+d\left(x,y\right)\geq d\left(x,z\right)$ * The identity morphism $0\to \operatorname\left(X,X\right)$ will be the unique morphism given from the fact that $0\geq d\left(X,X\right)$. * Since $R^*$ is a poset, all Diagram (category theory), diagrams that are required for an enriched category commute automatically. See the paper by F.W. Lawvere listed below.

* Assouad-Nagata dimension * * * * * * * * * * * * * * * * * * * * * *