![Manhattan distance](https://upload.wikimedia.org/wikipedia/commons/0/08/Manhattan_distance.svg)
In
mathematics, a metric space is a
set together with a notion of ''
distance
Distance is a numerical or occasionally qualitative measurement of how far apart objects or points are. In physics or everyday usage, distance may refer to a physical length or an estimation based on other criteria (e.g. "two counties over"). ...
'' between its
elements, usually called
points. The distance is measured by a
function
Function or functionality may refer to:
Computing
* Function key, a type of key on computer keyboards
* Function model, a structured representation of processes in a system
* Function object or functor or functionoid, a concept of object-oriente ...
called a metric or distance function. Metric spaces are the most general setting for studying many of the concepts of
mathematical analysis
Analysis is the branch of mathematics dealing with continuous functions, limit (mathematics), limits, and related theories, such as Derivative, differentiation, Integral, integration, measure (mathematics), measure, infinite sequences, series (m ...
and
geometry
Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is ...
.
The most familiar example of a metric space is
3-dimensional Euclidean space
Three-dimensional space (also: 3D space, 3-space or, rarely, tri-dimensional space) is a geometric setting in which three values (called ''parameters'') are required to determine the position of an element (i.e., point). This is the informa ...
with its usual notion of distance. Other well-known examples are a
sphere
A sphere () is a geometrical object that is a three-dimensional analogue to a two-dimensional circle. A sphere is the set of points that are all at the same distance from a given point in three-dimensional space.. That given point is th ...
equipped with the
angular distance
Angular distance \theta (also known as angular separation, apparent distance, or apparent separation) is the angle between the two sightlines, or between two point objects as viewed from an observer.
Angular distance appears in mathematics (in par ...
and the
hyperbolic plane
In mathematics, hyperbolic geometry (also called Lobachevskian geometry or Bolyai– Lobachevskian geometry) is a non-Euclidean geometry. The parallel postulate of Euclidean geometry is replaced with:
:For any given line ''R'' and point ''P' ...
. A metric may correspond to a metaphorical, rather than physical, notion of distance: for example, the set of 100-character Unicode strings can be equipped with the
Hamming distance
In information theory, the Hamming distance between two strings of equal length is the number of positions at which the corresponding symbols are different. In other words, it measures the minimum number of ''substitutions'' required to chan ...
, which measures the number of characters that need to be changed to get from one string to another.
Since they are very general, metric spaces are a tool used in many different branches of mathematics. Many types of mathematical objects have a natural notion of distance and therefore admit the structure of a metric space, including
Riemannian manifolds,
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, and
graph
Graph may refer to:
Mathematics
*Graph (discrete mathematics), a structure made of vertices and edges
**Graph theory, the study of such graphs and their properties
*Graph (topology), a topological space resembling a graph in the sense of discre ...
s. In
abstract algebra
In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures. Algebraic structures include group (mathematics), groups, ring (mathematics), rings, field (mathematics), fields, module (mathe ...
, the
''p''-adic numbers arise as elements of the
completion of a metric structure on the
rational numbers
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 rat ...
. Metric spaces are also studied in their own right in metric geometry and analysis on metric spaces.
Many of the basic notions of
mathematical analysis
Analysis is the branch of mathematics dealing with continuous functions, limit (mathematics), limits, and related theories, such as Derivative, differentiation, Integral, integration, measure (mathematics), measure, infinite sequences, series (m ...
, including
balls,
completeness, as well as
uniform
A uniform is a variety of clothing worn by members of an organization while participating in that organization's activity. Modern uniforms are most often worn by armed forces and paramilitary organizations such as police, emergency services, ...
,
Lipschitz, and
Hölder continuity Hölder:
* ''Hölder, Hoelder'' as surname
* Hölder condition
* Hölder's inequality
* Hölder mean
* Jordan–Hölder theorem In abstract algebra, a composition series provides a way to break up an algebraic structure, such as a group or a modu ...
, can be defined in the setting of metric spaces. Other notions, such as
continuity,
compactness, and
open
Open or OPEN may refer to:
Music
* Open (band), Australian pop/rock band
* The Open (band), English indie rock band
* ''Open'' (Blues Image album), 1969
* ''Open'' (Gotthard album), 1999
* ''Open'' (Cowboy Junkies album), 2001
* ''Open'' ( ...
and
closed sets, can be defined for metric spaces, but also in the even more general setting of
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 ...
s.
Definition and illustration
Motivation
![Great-circle distance vs straight line distance](https://upload.wikimedia.org/wikipedia/commons/5/5f/Great-circle_distance_vs_straight_line_distance.svg)
To see the utility of different notions of distance, consider the
surface of the Earth as a set of points. We can measure the distance between two such points by the length of the
shortest path along the surface, "
as the crow flies
__NOTOC__
The expression ''as the crow flies'' is an idiom for the most direct path between two points, rather similar to "in a beeline". This meaning is attested from the early 19th century, and appeared in Charles Dickens's 1838 novel ''Oliver ...
"; this is particularly useful for shipping and aviation. We can also measure the straight-line distance between two points through the Earth's interior; this notion is, for example, natural in
seismology
Seismology (; from Ancient Greek σεισμός (''seismós'') meaning "earthquake" and -λογία (''-logía'') meaning "study of") is the scientific study of earthquakes and the propagation of elastic waves through the Earth or through other ...
, since it roughly corresponds to the length of time it takes for seismic waves to travel between those two points.
The notion of distance encoded by the metric space axioms has relatively few requirements. This generality gives metric spaces a lot of flexibility. At the same time, the notion is strong enough to encode many intuitive facts about what distance means. This means that general results about metric spaces can be applied in many different contexts.
Like many fundamental mathematical concepts, the metric on a metric space can be interpreted in many different ways. A particular metric may not be best thought of as measuring physical distance, but, instead, as the cost of changing from one state to another (as with
Wasserstein metric
In mathematics, the Leonid Vaseršteĭn, Wasserstein distance or Leonid Kantorovich, Kantorovich–Gennadii Rubinstein, Rubinstein metric is a metric (mathematics), distance function defined between Probability distribution, probability distributi ...
s on spaces of
measures) or the degree of difference between two objects (for example, the
Hamming distance
In information theory, the Hamming distance between two strings of equal length is the number of positions at which the corresponding symbols are different. In other words, it measures the minimum number of ''substitutions'' required to chan ...
between two strings of characters, or the
Gromov–Hausdorff distance between metric spaces themselves).
Definition
Formally, a metric space is an
ordered pair where is a set and is a metric on , i.e., a
function
Function or functionality may refer to:
Computing
* Function key, a type of key on computer keyboards
* Function model, a structured representation of processes in a system
* Function object or functor or functionoid, a concept of object-oriente ...
satisfying the following axioms for all points
:
# The distance from a point to itself is zero:
Intuitively, it never costs anything to travel from a point to itself.
# (Positivity) The distance between two distinct points is always positive:
# (
Symmetry) The distance from to is always the same as the distance from to :
This excludes asymmetric notions of "cost" which arise naturally from the observation that it's harder to walk uphill than downhill.
# The
triangle inequality
In mathematics, the triangle inequality states that for any triangle, the sum of the lengths of any two sides must be greater than or equal to the length of the remaining side.
This statement permits the inclusion of degenerate triangles, but ...
holds:
This is a natural property of both physical and metaphorical notions of distance: you can arrive at from by taking a detour through , but this will not make your journey any faster than the shortest path.
If the metric is unambiguous, one often refers by
abuse of notation
In mathematics, abuse of notation occurs when an author uses a mathematical notation in a way that is not entirely formally correct, but which might help simplify the exposition or suggest the correct intuition (while possibly minimizing errors a ...
to "the metric space ".
Simple examples
The real numbers
The
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. Every ...
s with the distance function
given by the
absolute difference
The absolute difference of two real numbers x and y is given by , x-y, , the absolute value of their difference. It describes the distance on the real line between the points corresponding to x and y. It is a special case of the Lp distance for ...
form a metric space. Many properties of metric spaces and functions between them are generalizations of concepts in
real analysis
In mathematics, the branch of real analysis studies the behavior of real numbers, sequences and series of real numbers, and real functions. Some particular properties of real-valued sequences and functions that real analysis studies include conv ...
and coincide with those concepts when applied to the real line.
Metrics on Euclidean spaces
The Euclidean plane
can be equipped with many different metrics. The
Euclidean distance
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, therefor ...
familiar from school mathematics can be defined by
The
''taxicab'' or ''Manhattan'' distance is defined by
and can be thought of as the distance you need to travel along horizontal and vertical lines to get from one point to the other, as illustrated at the top of the article.
The ''maximum'',
, or ''
Chebyshev distance
In mathematics, Chebyshev distance (or Tchebychev distance), maximum metric, or L∞ metric is a metric defined on a vector space where the distance between two vectors is the greatest of their differences along any coordinate dimension. It is n ...
'' is defined by
This distance doesn't have an easy explanation in terms of paths in the plane, but it still satisfies the metric space axioms.
In fact, these three distances, while they have distinct properties, are similar in some ways. Informally, points that are close in one are close in the others, too. This observation can be quantified with the formula
which holds for every pair of points
.
A radically different distance can be defined by setting
In this ''discrete metric'', all distinct points are 1 unit apart: none of them are close to each other, and none of them are very far away from each other either. Intuitively, the discrete metric no longer remembers that the set is a plane, but treats it just as an undifferentiated set of points.
All of these metrics make sense on
as well as
.
Subspaces
Given a metric space and a
subset , we can consider to be a metric space by measuring distances the same way we would in . Formally, the ''induced metric'' on is a function
defined by
For example, if we take the two-dimensional sphere as a subset of
, the Euclidean metric on
induces the straight-line metric on described above. Two more useful examples are the open interval and the closed interval thought of as subspaces of the real line.
History
In 1906
Maurice Fréchet Maurice may refer to:
People
*Saint Maurice (died 287), Roman legionary and Christian martyr
*Maurice (emperor) or Flavius Mauricius Tiberius Augustus (539–602), Byzantine emperor
* Maurice (bishop of London) (died 1107), Lord Chancellor and L ...
introduced metric spaces in his work ''Sur quelques points du calcul fonctionnel'' in the context of
functional analysis
Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (e.g. inner product, norm, topology, etc.) and the linear functions defined o ...
: his main interest was in studying the real-valued functions from a metric space, generalizing the theory of functions of several or even infinitely many variables, as pioneered by mathematicians such as
Cesare Arzelà
Cesare Arzelà (6 March 1847 – 15 March 1912) was an Italian mathematician who taught at the University of Bologna and is recognized for his contributions in the theory of functions, particularly for his characterization of sequences of continu ...
. The idea was further developed and placed in its proper context by
Felix Hausdorff
Felix Hausdorff ( , ; November 8, 1868 – January 26, 1942) was a German mathematician who is considered to be one of the founders of modern topology and who contributed significantly to set theory, descriptive set theory, measure theory, an ...
in his magnum opus
''Principles of Set Theory'', which also introduced the notion of a
(Hausdorff) topological space.
General metric spaces have become a foundational part of the mathematical curriculum. Prominent examples of metric spaces in mathematical research include Riemannian manifolds and normed vector spaces, which are the domain of
differential geometry and
functional analysis
Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (e.g. inner product, norm, topology, etc.) and the linear functions defined o ...
, respectively.
Fractal geometry
In mathematics, a fractal is a geometric shape containing detailed structure at arbitrarily small scales, usually having a fractal dimension strictly exceeding the topological dimension. Many fractals appear similar at various scales, as illu ...
is a source of some exotic metric spaces. Others have arisen as limits through the study of discrete or smooth objects, including
scale-invariant limits in
statistical physics
Statistical physics is a branch of physics that evolved from a foundation of statistical mechanics, which uses methods of probability theory and statistics, and particularly the mathematical tools for dealing with large populations and approxim ...
,
Alexandrov space In geometry, Alexandrov spaces with curvature ≥ ''k'' form a generalization of Riemannian manifolds with sectional curvature ≥ ''k'', where ''k'' is some real number. By definition, these spaces are locally compact complete length spaces where t ...
s arising as
Gromov–Hausdorff limits of sequences of Riemannian manifolds, and
boundaries and
asymptotic cone
In mathematics, an ultralimit is a geometric construction that assigns to a sequence of metric spaces ''Xn'' a limiting metric space. The notion of an ultralimit captures the limiting behavior of finite configurations in the spaces ''Xn'' and use ...
s in
geometric group theory
Geometric group theory is an area in mathematics devoted to the study of finitely generated groups via exploring the connections between algebraic properties of such group (mathematics), groups and topology, topological and geometry, geometric pro ...
. Finally, many new applications of finite and discrete metric spaces have arisen in
computer science
Computer science is the study of computation, automation, and information. Computer science spans theoretical disciplines (such as algorithms, theory of computation, information theory, and automation) to practical disciplines (includi ...
.
Basic notions
A distance function is enough to define notions of closeness and convergence that were first developed in
real analysis
In mathematics, the branch of real analysis studies the behavior of real numbers, sequences and series of real numbers, and real functions. Some particular properties of real-valued sequences and functions that real analysis studies include conv ...
. Properties that depend on the structure of a metric space are referred to as ''metric properties''. Every metric space is also 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 some metric properties can also be rephrased without reference to distance in the language of topology; that is, they are really
topological properties
In topology and related areas of mathematics, a topological property or topological invariant is a property of a topological space that is invariant under homeomorphisms. Alternatively, a topological property is a proper class of topological space ...
.
The topology of a metric space
For any point in a metric space and any real number , the
''open ball'' of radius around is defined to be the set of points that are at most distance from :
This is a natural way define a set of points that are relatively close to . Therefore, a set
is a
''neighborhood'' of (informally, it contains all points "close enough" to ) if it contains an open ball of radius around for some .
An ''open set'' is a set which is a neighborhood of all its points. It follows that the open balls form a
base for a topology on . In other words, the open sets of are exactly the unions of open balls. As in any topology,
closed sets 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 , , and defined above all induce the same topology on
, although they behave differently in many respects. Similarly,
with the Euclidean metric and its subspace the interval with the induced metric are
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'' and are particularly well-behaved in many ways: in particular, they are
paracompact
In mathematics, a paracompact space is a topological space in which every open cover has an open refinement that is locally finite. These spaces were introduced by . Every compact space is paracompact. Every paracompact Hausdorff space is normal ...
Hausdorff space
In topology and related branches of mathematics, a Hausdorff space ( , ), separated space or T2 space is a topological space where, for any two distinct points, there exist neighbourhoods of each which are disjoint from each other. Of the m ...
s (hence
normal Normal(s) or The Normal(s) may refer to:
Film and television
* ''Normal'' (2003 film), starring Jessica Lange and Tom Wilkinson
* ''Normal'' (2007 film), starring Carrie-Anne Moss, Kevin Zegers, Callum Keith Rennie, and Andrew Airlie
* ''Norma ...
) and
first-countable
In topology, a branch of mathematics, a first-countable space is a topological space satisfying the "first axiom of countability". Specifically, a space X is said to be first-countable if each point has a countable neighbourhood basis (local base) ...
. The
Nagata–Smirnov metrization theorem
The Nagata–Smirnov metrization theorem in topology characterizes when a topological space is metrizable. The theorem states that a topological space X is metrizable if and only if it is regular, Hausdorff and has a countably locally finite (t ...
gives a characterization of metrizability in terms of other topological properties, without reference to metrics.
Convergence
Convergence of sequences in Euclidean space is defined as follows:
: A sequence converges to a point if for every there is an integer such that for all , .
Convergence of sequences in a topological space is defined as follows:
: A sequence converges to a point if for every open set containing there is an integer such that for all ,
.
In metric spaces, both of these definitions make sense and they are equivalent. This is a general pattern for
topological properties
In topology and related areas of mathematics, a topological property or topological invariant is a property of a topological space that is invariant under homeomorphisms. Alternatively, a topological property is a proper class of topological space ...
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
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 in a metric space is
''Cauchy'' if for every there is an integer such that for all , . By the triangle inequality, any convergent sequence is Cauchy: if and are both less than away from the limit, then they are less than away from each other. If the converse is true—every Cauchy sequence in converges—then is complete.
Euclidean spaces are complete, as is
with the other metrics described above. Two examples of spaces which are not complete are and the rationals, each with the metric induced from
. One can think of 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
(for example, its successive decimal approximations). These examples show that completeness is ''not'' a topological property, since
is complete but the homeomorphic space is not.
This notion of "missing points" can be made precise. In fact, every metric space has a unique
''completion'', which is a complete space that contains the given space as a
dense
Density (volumetric mass density or specific mass) is the substance's mass per unit of volume. The symbol most often used for density is ''ρ'' (the lower case Greek letter rho), although the Latin letter ''D'' can also be used. Mathematically ...
subset. For example, is the completion of , 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 are defined as the completion of the rationals under a different metric. Completion is particularly common as a tool in
functional analysis
Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (e.g. inner product, norm, topology, etc.) and the linear functions defined o ...
. 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
In mathematics, a weak solution (also called a generalized solution) to an ordinary or partial differential equation is a function for which the derivatives may not all exist but which is nonetheless deemed to satisfy the equation in some precis ...
s to
differential equation
In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, an ...
s typically live in a completion (a
Sobolev space
In mathematics, a Sobolev space is a vector space of functions equipped with a norm that is a combination of ''Lp''-norms of the function together with its derivatives up to a given order. The derivatives are understood in a suitable weak sense ...
) rather than the original space of nice functions for which the differential equation actually makes sense.
Bounded and totally bounded spaces
![Diameter of a Set](https://upload.wikimedia.org/wikipedia/commons/f/fc/Diameter_of_a_Set.svg)
A metric space is ''bounded'' if there is an such that no pair of points in is more than distance apart. The least such is called the ' of .
The space is called ''precompact'' or ''
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 ...
'' if for every there is a finite
cover of by open balls of radius . Every totally bounded space is bounded. To see this, start with a finite cover by -balls for some arbitrary . Since the subset of consisting of the centers of these balls is finite, it has finite diameter, say . By the triangle inequality, the diameter of the whole space is at most . The converse does not hold: an example of a metric space that is bounded but not totally bounded is
(or any other infinite set) with the discrete metric.
Compactness
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 is compact if every open cover has a finite subcover (the usual topological definition).
# A metric space is compact if every sequence has a convergent subsequence. (For general topological spaces this is called
sequential compactness and is not equivalent to compactness.)
# A metric space is compact if it is complete and totally bounded. (This definition is written in terms of metric properties and doesn't 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 .
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.
Functions between metric spaces
![Functions between metric spaces](https://upload.wikimedia.org/wikipedia/commons/9/9c/Functions_between_metric_spaces.svg)
Unlike in the case of topological spaces or algebraic structures such as
group
A group is a number of persons or things that are located, gathered, or classed together.
Groups of people
* Cultural group, a group whose members share the same cultural identity
* Ethnic group, a group whose members share the same ethnic ide ...
s or
ring
Ring may refer to:
* Ring (jewellery), a round band, usually made of metal, worn as ornamental jewelry
* To make a sound with a bell, and the sound made by a bell
:(hence) to initiate a telephone connection
Arts, entertainment and media Film and ...
s, there is no single "right" type of
structure-preserving function between metric spaces. Instead, one works with different types of functions depending on one's goals. Throughout this section, suppose that
and
are two metric spaces. The words "function" and "map" are used interchangeably.
Isometries
One interpretation of a "structure-preserving" map is one that fully preserves the distance function:
: A function
is ''distance-preserving'' if for every pair of points and in ,
It follows from the metric space axioms that a distance-preserving function is injective. A bijective distance-preserving function is called an ''isometry''. One perhaps non-obvious example of an isometry between spaces described in this article is the map
defined by
If there is an isometry between the spaces and , they are said to be ''isometric''. Metric spaces that are isometric are
essentially identical.
Continuous maps
On the other end of the spectrum, one can forget entirely about the metric structure and study
continuous maps
In mathematics, a continuous function is a function such that a continuous variation (that is a change without jump) of the argument induces a continuous variation of the value of the function. This means that there are no abrupt changes in va ...
, which only preserve topological structure. There are several equivalent definitions of continuity for metric spaces. The most important are:
* Topological definition. A function
is continuous if for every open set in , the
preimage is open.
*
Sequential continuity
In mathematics, a continuous function is a function such that a continuous variation (that is a change without jump) of the argument induces a continuous variation of the value of the function. This means that there are no abrupt changes in va ...
. A function
is continuous if whenever a sequence converges to a point in , the sequence
converges to the point in .
: (These first two definitions are ''not'' equivalent for all topological spaces.)
* ε–δ definition. A function
is continuous if for every point in and every there exists such that for all in we have
A ''
homeomorphism
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 isomor ...
'' is a continuous map whose inverse is also continuous; if there is a homeomorphism between and , they are said to be ''homeomorphic''. Homeomorphic spaces are the same from the point of view of topology, but may have very different metric properties. For example,
is unbounded and complete, while is bounded but not complete.
Uniformly continuous maps
A function
is ''
uniformly continuous
In mathematics, a real function f of real numbers is said to be uniformly continuous if there is a positive real number \delta such that function values over any function domain interval of the size \delta are as close to each other as we want. In ...
'' if for every real number there exists such that for all points and in such that
, we have
The only difference between this definition and the ε–δ definition of continuity is the order of quantifiers: the choice of δ must depend only on ε and not on the point . However, this subtle change makes a big difference. For example, uniformly continuous maps take Cauchy sequences in to Cauchy sequences in . This implies that the image of a complete space under a uniformly continuous map is complete. In other words, uniform continuity preserves some metric properties which are not purely topological.
On the other hand, the
Heine–Cantor theorem
In mathematics, the Heine–Cantor theorem, named after Eduard Heine and Georg Cantor, states that if f \colon M \to N is a continuous function between two metric spaces M and N, and M is compact, then f is uniformly continuous. An important speci ...
states that if is compact, then every continuous map is uniformly continuous. In other words, uniform continuity cannot distinguish any non-topological features of compact metric spaces.
Lipschitz maps and contractions
A
Lipschitz map
In mathematical analysis, Lipschitz continuity, named after German mathematician Rudolf Lipschitz, is a strong form of uniform continuity for functions. Intuitively, a Lipschitz continuous function is limited in how fast it can change: there exis ...
is one that stretches distances by at most a bounded factor. Formally, given a real number , the map
is -''Lipschitz'' if
Lipschitz maps are particularly important in metric geometry, since they provide more flexibility than distance-preserving maps, but still make essential use of the metric. For example, a curve in a metric space is
rectifiable (has finite length) if and only if it has a Lipschitz reparametrization.
A 1-Lipschitz map is sometimes called a ''nonexpanding'' or ''
metric map In the mathematical theory of metric spaces, a metric map is a function between metric spaces that does not increase any distance (such functions are always continuous).
These maps are the morphisms in the category of metric spaces, Met (Isbell 1 ...
''. Metric maps are commonly taken to be the morphisms of the
category of metric spaces In category theory, Met is a category that has metric spaces as its objects and metric maps (continuous functions between metric spaces that do not increase any pairwise distance) as its morphisms. This is a category because the composition of two ...
.
A -Lipschitz map for is called a ''
contraction''. 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 certa ...
states that if is a complete metric space, then every contraction
admits a unique
fixed point. If the metric space is compact, the result holds for a slightly weaker condition on : a map
admits a unique fixed point if
Quasi-isometries
A
quasi-isometry In mathematics, a quasi-isometry is a function between two metric spaces that respects large-scale geometry of these spaces and ignores their small-scale details. Two metric spaces are quasi-isometric if there exists a quasi-isometry between them. ...
is a map that preserves the "large-scale structure" of a metric space. Quasi-isometries need not be continuous. For example,
and its subspace
are quasi-isometric, even though one is connected and the other is discrete. The equivalence relation of quasi-isometry is important in
geometric group theory
Geometric group theory is an area in mathematics devoted to the study of finitely generated groups via exploring the connections between algebraic properties of such group (mathematics), groups and topology, topological and geometry, geometric pro ...
: the
Švarc–Milnor lemma In the mathematical subject of geometric group theory, the Švarc–Milnor lemma (sometimes also called Milnor–Švarc lemma, with both variants also sometimes spelling Švarc as Schwarz) is a statement which says that a group G, equipped with ...
states that all spaces on which a group
acts geometrically are quasi-isometric.
Formally, the map
is a ''quasi-isometric embedding'' if there exist constants and such that
It is a ''quasi-isometry'' if in addition it is ''quasi-surjective'', i.e. there is a constant such that every point in
is at distance at most from some point in the image
.
Notions of metric space equivalence
Given two metric spaces
and
:
*They are called homeomorphic (topologically isomorphic) if there is a
homeomorphism
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 isomor ...
between them (i.e., a continuous
bijection with a continuous inverse). If
and the identity map is a homeomorphism, then
and
are said to be topologically equivalent.
*They are called uniformic (uniformly isomorphic) if there is a
uniform isomorphism between them (i.e., a uniformly continuous bijection with a uniformly continuous inverse).
*They are called bilipschitz homeomorphic if there is a bilipschitz bijection between them (i.e., a Lipschitz bijection with a Lipschitz inverse).
*They are called isometric if there is a (bijective)
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'' me ...
between them. In this case, the two metric spaces are essentially identical.
*They are called quasi-isometric if there is a
quasi-isometry In mathematics, a quasi-isometry is a function between two metric spaces that respects large-scale geometry of these spaces and ignores their small-scale details. Two metric spaces are quasi-isometric if there exists a quasi-isometry between them. ...
between them.
Metric spaces with additional structure
Normed vector spaces
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 ...
is a vector space equipped with a ''
norm
Naturally occurring radioactive materials (NORM) and technologically enhanced naturally occurring radioactive materials (TENORM) consist of materials, usually industrial wastes or by-products enriched with radioactive elements found in the envi ...
'', which is a function that measures the length of vectors. The norm of a vector is typically denoted by
. Any normed vector space can be equipped with a metric in which the distance between two vectors and is given by
The metric is said to be ''induced'' by the norm
. Conversely, if a metric on a
vector space
In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called ''scalars''. Scalars are often real numbers, but can ...
is
* translation invariant:
for every , , and in ; and
*
:
for every and in and real number ;
then it is the metric induced by the norm
A similar relationship holds between
seminorm In mathematics, particularly in functional analysis, a seminorm is a vector space norm that need not be positive definite. Seminorms are intimately connected with convex sets: every seminorm is the Minkowski functional of some absorbing disk ...
s and
pseudometrics.
Among examples of metrics induced by a norm are the metrics , , and on
, which are induced by the
Manhattan norm
A taxicab geometry or a Manhattan geometry is a geometry whose 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 ...
, the
Euclidean norm, and the
maximum 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 the ...
, respectively. More generally, the
Kuratowski embedding allows one to see any metric space as a subspace of a normed vector space.
Infinite-dimensional normed vector spaces, particularly spaces of functions, are studied in
functional analysis
Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (e.g. inner product, norm, topology, etc.) and the linear functions defined o ...
. Completeness is particularly important in this context: a complete normed vector space is known as a
Banach space. An unusual property of normed vector spaces is that
linear transformation
In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping V \to W between two vector spaces that pre ...
s between them are continuous if and only if they are Lipschitz. Such transformations are known as
bounded operator
In functional analysis and operator theory, a bounded linear operator is a linear transformation L : X \to Y between topological vector spaces (TVSs) X and Y that maps bounded subsets of X to bounded subsets of Y.
If X and Y are normed vector ...
s.
Length spaces
![approximate arc length](https://upload.wikimedia.org/wikipedia/commons/1/11/Approximate_arc_length.svg)
A
curve
In mathematics, a curve (also called a curved line in older texts) is an object similar to a line, but that does not have to be straight.
Intuitively, a curve may be thought of as the trace left by a moving point. This is the definition that ...
in a metric space is a continuous function
. The
length of is measured by
In general, this supremum may be infinite; a curve of finite length is called ''rectifiable''. Suppose that the length of the curve is equal to the distance between its endpoints—that is, it's the shortest possible path between its endpoints. After reparametrization by arc length, becomes a ''
geodesic'': a curve which is a distance-preserving function. A geodesic is a shortest possible path between any two of its points.
A ''geodesic metric space'' is a metric space which admits a geodesic between any two of its points. The spaces
and
are both geodesic metric spaces. In
, geodesics are unique, but in
, there are often infinitely many geodesics between two points, as shown in the figure at the top of the article.
The space is a ''
length space In the mathematical study of metric spaces, one can consider the arclength of paths in the space. If two points are at a given distance from each other, it is natural to expect that one should be able to get from the first point to the second alo ...
'' (or the metric is ''intrinsic'') if the distance between any two points and is the infimum of lengths of paths between them. Unlike in a geodesic metric space, the infimum does not have to be attained. An example of a length space which is not geodesic is the Euclidean plane minus the origin: the points and can be joined by paths of length arbitrarily close to 2, but not by a path of length 2. An example of a metric space which is not a length space is given by the straight-line metric on the sphere: the straight line between two points through the center of the earth is shorter than any path along the surface.
Given any metric space , one can define a new, intrinsic distance function on by setting the distance between points and to be infimum of the -lengths of paths between them. For instance, if is the straight-line distance on the sphere, then is the great-circle distance. However, in some cases may have infinite values. For example, if is the
Koch snowflake
The Koch snowflake (also known as the Koch curve, Koch star, or Koch island) is a fractal curve and one of the earliest fractals to have been described. It is based on the Koch curve, which appeared in a 1904 paper titled "On a Continuous Curv ...
with the subspace metric induced from
, then the resulting intrinsic distance is infinite for any pair of distinct points.
Riemannian manifolds
A
Riemannian manifold is a space equipped with a Riemannian
metric tensor, which determines lengths of
tangent vectors
In mathematics, a tangent vector is a vector that is tangent to a curve or surface at a given point. Tangent vectors are described in the differential geometry of curves in the context of curves in R''n''. More generally, tangent vectors are eleme ...
at every point. This can be thought of defining a notion of distance infinitesimally. In particular, a differentiable path
in a Riemannian manifold has length defined as the integral of the length of the tangent vector to the path:
On a connected Riemannian manifold, one then defines the distance between two points as the infimum of lengths of smooth paths between them. This construction generalizes to other kinds of infinitesimal metrics on manifolds, such as
sub-Riemannian and
Finsler metrics.
The Riemannian metric is uniquely determined by the distance function; this means that in principle, all information about a Riemannian manifold can be recovered from its distance function. One direction in metric geometry is finding purely metric (
"synthetic") formulations of properties of Riemannian manifolds. For example, a Riemannian manifold is a
space (a synthetic condition which depends purely on the metric) if and only if its
sectional curvature In Riemannian geometry, the sectional curvature is one of the ways to describe the curvature of Riemannian manifolds. The sectional curvature ''K''(σ''p'') depends on a two-dimensional linear subspace σ''p'' of the tangent space at a poi ...
is bounded above by . Thus spaces generalize upper curvature bounds to general metric spaces.
Metric measure spaces
Real analysis makes use of both the metric on
and the
Lebesgue measure. Therefore, generalizations of many ideas from analysis naturally reside in
metric measure spaces: spaces that have both a
measure and a metric which are compatible with each other. Formally, a ''metric measure space'' is a metric space equipped with a
Borel regular measure Borel may refer to:
People
* Borel (author), 18th-century French playwright
* Borel (1906–1967), pseudonym of the French actor Jacques Henri Cottance
* Émile Borel (1871 – 1956), a French mathematician known for his founding work in the are ...
such that every ball has positive measure. For example Euclidean spaces of dimension , and more generally -dimensional Riemannian manifolds, naturally have the structure of a metric measure space, equipped with the
Lebesgue measure. Certain
fractal metric spaces such as the
Sierpiński gasket can be equipped with the α-dimensional
Hausdorff measure
In mathematics, Hausdorff measure is a generalization of the traditional notions of area and volume to non-integer dimensions, specifically fractals and their Hausdorff dimensions. It is a type of outer measure, named for Felix Hausdorff, that as ...
where α is the
Hausdorff dimension
In mathematics, Hausdorff dimension is a measure of ''roughness'', or more specifically, fractal dimension, that was first introduced in 1918 by mathematician Felix Hausdorff. For instance, the Hausdorff dimension of a single point is zero, of ...
. In general, however, a metric space may not have an "obvious" choice of measure.
One application of metric measure spaces is generalizing the notion of
Ricci curvature
In differential geometry, the Ricci curvature tensor, named after Gregorio Ricci-Curbastro, is a geometric object which is determined by a choice of Riemannian or pseudo-Riemannian metric on a manifold. It can be considered, broadly, as a measur ...
beyond Riemannian manifolds. Just as and
Alexandrov space In geometry, Alexandrov spaces with curvature ≥ ''k'' form a generalization of Riemannian manifolds with sectional curvature ≥ ''k'', where ''k'' is some real number. By definition, these spaces are locally compact complete length spaces where t ...
s generalize scalar curvature bounds,
RCD space RCD may refer to:
Science and technology
* Residual-current device, a safety device that breaks an electrical circuit
* Rabbit calicivirus disease, a disease caused by the Rabbit haemorrhagic disease virus
* Registered clock driver chips, part of ...
s are a class of metric measure spaces which generalize lower bounds on Ricci curvature.
Further examples and applications
Graphs and finite metric spaces
A if its induced topology is the
discrete topology
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 top ...
. Although many concepts, such as completeness and compactness, are not interesting for such spaces, they are nevertheless an object of study in several branches of mathematics. In particular, (those having a
finite
Finite is the opposite of 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 or marke ...
number of points) are studied in
combinatorics and
theoretical computer science
computer science (TCS) is a subset of general computer science and mathematics that focuses on mathematical aspects of computer science such as the theory of computation, lambda calculus, and type theory.
It is difficult to circumscribe the ...
. Embeddings in other metric spaces are particularly well-studied. For example, not every finite metric space can be
isometrically embedded in a Euclidean space or in
Hilbert space. On the other hand, in the worst case the required distortion (bilipschitz constant) is only logarithmic in the number of points.
For any
undirected connected graph , the set of vertices of can be turned into a metric space by defining the
distance
Distance is a numerical or occasionally qualitative measurement of how far apart objects or points are. In physics or everyday usage, distance may refer to a physical length or an estimation based on other criteria (e.g. "two counties over"). ...
between vertices and to be the length of the shortest edge path connecting them. This is also called ''shortest-path distance'' or ''geodesic distance''. In
geometric group theory
Geometric group theory is an area in mathematics devoted to the study of finitely generated groups via exploring the connections between algebraic properties of such group (mathematics), groups and topology, topological and geometry, geometric pro ...
this construction is applied to the
Cayley graph
In mathematics, a Cayley graph, also known as a Cayley color graph, Cayley diagram, group diagram, or color group is a graph that encodes the abstract structure of a group. Its definition is suggested by Cayley's theorem (named after Arthur Cay ...
of a (typically infinite)
finitely-generated group, yielding the
word metric In group theory, a word metric on a discrete group G is a way to measure distance between any two elements of G . As the name suggests, the word metric is a metric on G , assigning to any two elements g , h of G a distance d(g,h) that m ...
. Up to a bilipschitz homeomorphism, the word metric depends only on the group and not on the chosen finite generating set.
Distances between mathematical objects
In modern mathematics, one often studies spaces whose points are themselves mathematical objects. A distance function on such a space generally aims to measure the dissimilarity between two objects. Here are some examples:
* Functions to a metric space. If is any set and is a metric space, then the set of all
bounded functions
(i.e. those functions whose image is a
bounded subset of
) can be turned into a metric space by defining the distance between two bounded functions and to be
This metric is called the
uniform metric
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 ...
or supremum metric. If is complete, then this
function space is complete as well; moreover, if is also a topological space, then the subspace consisting 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 ...
functions from to is also complete. When is a subspace of
, this function space is known as a
classical Wiener space
In mathematics, classical Wiener space is the collection of all continuous functions on a given domain (usually a subinterval of the real line), taking values in a metric space (usually ''n''-dimensional Euclidean space). Classical Wiener space i ...
.
*
String metric
In mathematics and computer science, a string metric (also known as a string similarity metric or string distance function) is a metric that measures distance ("inverse similarity") between two text strings for approximate string matching or comp ...
s and
edit distance
In computational linguistics and computer science, edit distance is a string metric, i.e. a way of quantifying how dissimilar two strings (e.g., words) are to one another, that is measured by counting the minimum number of operations required to ...
s. There are many ways of measuring distances between
strings of characters, which may represent
sentences in
computational linguistics or
code word
In communication, a code word is an element of a standardized code or protocol. Each code word is assembled in accordance with the specific rules of the code and assigned a unique meaning. Code words are typically used for reasons of reliability, ...
s in
coding theory
Coding theory is the study of the properties of codes and their respective fitness for specific applications. Codes are used for data compression, cryptography, error detection and correction, data transmission and data storage. Codes are studied ...
. ''Edit distances'' attempt to measure the number of changes necessary to get from one string to another. For example, the
Hamming distance
In information theory, the Hamming distance between two strings of equal length is the number of positions at which the corresponding symbols are different. In other words, it measures the minimum number of ''substitutions'' required to chan ...
measures the minimal number of substitutions needed, while the
Levenshtein distance
In information theory, linguistics, and computer science, the Levenshtein distance is a string metric for measuring the difference between two sequences. Informally, the Levenshtein distance between two words is the minimum number of single-charac ...
measures the minimal number of deletions, insertions, and substitutions; both of these can be thought of as distances in an appropriate graph.
*
Graph edit distance In mathematics and computer science, graph edit distance (GED) is a measure of similarity (or dissimilarity) between two graphs.
The concept of graph edit distance was first formalized mathematically by Alberto Sanfeliu and King-Sun Fu in 1983.
A ...
is a measure of dissimilarity between two
graphs, defined as the minimal number of
graph edit operations required to transform one graph into another.
*
Wasserstein metric
In mathematics, the Leonid Vaseršteĭn, Wasserstein distance or Leonid Kantorovich, Kantorovich–Gennadii Rubinstein, Rubinstein metric is a metric (mathematics), distance function defined between Probability distribution, probability distributi ...
s measure the distance between two
measures on the same metric space. The Wasserstein distance between two measures is, roughly speaking, the
cost of transporting one to the other.
* The set of all by
matrices
Matrix most commonly refers to:
* ''The Matrix'' (franchise), an American media franchise
** ''The Matrix'', a 1999 science-fiction action film
** "The Matrix", a fictional setting, a virtual reality environment, within ''The Matrix'' (franchis ...
over some
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 ...
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, such as:
Level or position in a hierarchical organization
* Academic rank
* Diplomatic rank
* Hierarchy
* ...
distance
.
* The
Helly metric in
game theory measures the difference between
strategies
Strategy (from Greek στρατηγία ''stratēgia'', "art of troop leader; office of general, command, generalship") is a general plan to achieve one or more long-term or overall goals under conditions of uncertainty. In the sense of the " ar ...
in a game.
Hausdorff and Gromov–Hausdorff distance
The idea of spaces of mathematical objects can also be applied to subsets of a metric space, as well as metric spaces themselves.
Hausdorff and
Gromov–Hausdorff distance define metrics on the set of compact subsets of a metric space and the set of compact metric spaces, respectively.
Suppose is a metric space, and let be a subset of . The ''distance from to a point of '' is, informally, the distance from to the closest point of . However, since there may not be a single closest point, it is defined via an
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 each element of S, if such an element exists. Consequently, the term ''greatest lo ...
:
In particular,
if and only if belongs to the
closure of . Furthermore, distances between points and sets satisfy a version of the triangle inequality:
and therefore the map
defined by
is continuous. Incidentally, this shows that metric spaces are
completely regular
In topology and related branches of mathematics, Tychonoff spaces and completely regular spaces are kinds of topological spaces. These conditions are examples of separation axioms. A Tychonoff space refers to any completely regular space that is ...
.
Given two subsets and of , their ''Hausdorff distance'' is
Informally, two sets and are close to each other in the Hausdorff distance if no element of is too far from and vice versa. For example, if is an open set in Euclidean space is an
ε-net inside , then
. In general, the Hausdorff distance
can be infinite or zero. However, the Hausdorff distance between two distinct compact sets is always positive and finite. Thus the Hausdorff distance defines a metric on the set of compact subsets of .
The Gromov–Hausdorff metric defines a distance between (isometry classes of) compact metric spaces. The ''Gromov–Hausdorff distance'' between compact spaces and is the infimum of the Hausdorff distance over all metric spaces that contain and as subspaces. While the exact value of the Gromov–Hausdorff distance is rarely useful to know, the resulting topology has found many applications.
Miscellaneous examples
* Given a metric space and an increasing
concave function such that if and only if , then
is also a metric on . If for some real number , such a metric is known as a snowflake of .
* The tight span of a metric space is another metric space which can be thought of as an abstract version of the convex hull.
* The British Rail metric (also called the "post office metric" or the "SNCF metric") on 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 ...
is given by
for distinct points
and
, and
. More generally
can be replaced with a function
taking an arbitrary set
to non-negative reals and taking the value
at most once: then the metric is defined on
by
for distinct points
and
, and The name alludes to the tendency of railway journeys to proceed via London (or Paris) irrespective of their final destination.
* The
Robinson–Foulds metric used for calculating the distances between
Phylogenetic trees in
Phylogenetics
In biology, phylogenetics (; from Greek φυλή/ φῦλον [] "tribe, clan, race", and wikt:γενετικός, γενετικός [] "origin, source, birth") is the study of the evolutionary history and relationships among or within groups ...
Constructions
Product metric spaces
If
are metric spaces, and is the
Euclidean norm on
, then
is a metric space, where the
product metric is defined by
and the induced topology agrees with the
product topology
In topology and related areas of mathematics, a product space is the Cartesian product of a family of topological spaces equipped with a natural topology called the product topology. This topology differs from another, perhaps more natural-s ...
. By the equivalence of norms in finite dimensions, a topologically equivalent metric is obtained if is the
taxicab norm
A taxicab geometry or a Manhattan geometry is a geometry whose usual distance function or 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 their Cartesian co ...
, a
p-norm, the
maximum 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 the ...
, or any other norm which is non-decreasing as the coordinates of a positive -tuple increase (yielding the triangle inequality).
Similarly, a metric on the topological product of countably many metric spaces can be obtained using the metric
The topological product of uncountably many metric spaces need not be metrizable. For example, an uncountable product of copies of
is not
first-countable
In topology, a branch of mathematics, a first-countable space is a topological space satisfying the "first axiom of countability". Specifically, a space X is said to be first-countable if each point has a countable neighbourhood basis (local base) ...
and thus isn't metrizable.
Quotient metric spaces
If is a metric space with metric , and
is an
equivalence relation on , then we can endow the quotient set
with a pseudometric. The distance between two equivalence classes