Metric space

# Metric space

Discussion

Encyclopedia
In mathematics
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, a metric space is a set where a notion of distance
Distance
Distance is a numerical description of how far apart objects are. In physics or everyday discussion, distance may refer to a physical length, or an estimation based on other criteria . In mathematics, a distance function or metric is a generalization of the concept of physical distance...

(called a metric
Metric (mathematics)
In mathematics, a metric or distance function is a function which defines a distance between elements of a set. A set with a metric is called a metric space. A metric induces a topology on a set but not all topologies can be generated by a metric...

) between elements of the set is defined.

The metric space which most closely corresponds to our intuitive understanding of space is the 3-dimensional Euclidean space
Euclidean space
In mathematics, Euclidean space is the Euclidean plane and three-dimensional space of Euclidean geometry, as well as the generalizations of these notions to higher dimensions...

. In fact, the notion of "metric" is a generalization of the Euclidean metric 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 straight line segment
Line segment
In geometry, a line segment is a part of a line that is bounded by two end points, and contains every point on the line between its end points. Examples of line segments include the sides of a triangle or square. More generally, when the end points are both vertices of a polygon, the line segment...

connecting them.

The geometric properties of the space depend on the metric chosen, and by using a different metric we can construct interesting non-Euclidean geometries such as those used in the theory of general relativity
General relativity
General relativity or the general theory of relativity is the geometric theory of gravitation published by Albert Einstein in 1916. It is the current description of gravitation in modern physics...

.

A metric space also induces topological properties like open
Open set
The concept of an open set is fundamental to many areas of mathematics, especially point-set topology and metric topology. Intuitively speaking, a set U is open if any point x in U can be "moved" a small amount in any direction and still be in the set U...

and closed set
Closed set
In geometry, topology, and related branches of mathematics, a closed set is a set whose complement is an open set. In a topological space, a closed set can be defined as a set which contains all its limit points...

s which leads to the study of even more abstract topological space
Topological space
Topological spaces are mathematical structures that allow the formal definition of concepts such as convergence, connectedness, and continuity. They appear in virtually every branch of modern mathematics and are a central unifying notion...

s.

## History

Maurice Fréchet introduced metric spaces in his work Sur quelques points du calcul fonctionnel, Rendic. Circ. Mat. Palermo 22 (1906) 1–74.

## Definition

A metric space is an ordered pair
Ordered pair
In mathematics, an ordered pair is a pair of mathematical objects. In the ordered pair , the object a is called the first entry, and the object b the second entry of the pair...

where is a set and is a metric
Metric (mathematics)
In mathematics, a metric or distance function is a function which defines a distance between elements of a set. A set with a metric is called a metric space. A metric induces a topology on a set but not all topologies can be generated by a metric...

on , i.e., a function

such that for any , the following holds:
1.     (non-negative) ,
2. iff
If and only if
In logic and related fields such as mathematics and philosophy, if and only if is a biconditional logical connective between statements....

(identity of indiscernibles
Identity of indiscernibles
The identity of indiscernibles is an ontological principle which states that two or more objects or entities are identical if they have all their properties in common. That is, entities x and y are identical if any predicate possessed by x is also possessed by y and vice versa...

),
3.     (symmetry) and
4.     (triangle inequality
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 ....

) .

The first condition follows from the other three, since:

The function is also called distance function or simply distance. Often, is omitted and one just writes for a metric space if it is clear from the context what metric is used.

## Examples of metric spaces

• 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. Many of the examples below can be seen as concrete versions of this general idea.
• The real number
Real number
In mathematics, a real number is a value that represents a quantity along a continuum, such as -5 , 4/3 , 8.6 , √2 and π...

s with the distance function given by the absolute difference
Absolute difference
The absolute difference of two real numbers x, 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...

, and more generally Euclidean -space
Euclidean space
In mathematics, Euclidean space is the Euclidean plane and three-dimensional space of Euclidean geometry, as well as the generalizations of these notions to higher dimensions...

with the Euclidean distance
Euclidean distance
In mathematics, the Euclidean distance or Euclidean metric is the "ordinary" distance between two points that one would measure with a ruler, and is given by the Pythagorean formula. By using this formula as distance, Euclidean space becomes a metric space...

, are complete
Complete space
In mathematical analysis, a metric space M is called complete if every Cauchy sequence of points in M has a limit that is also in M or, alternatively, if every Cauchy sequence in M converges in M....

metric spaces. The rational number
Rational number
In mathematics, a rational number is any number that can be expressed as the quotient or fraction a/b of two integers, with the denominator b not equal to zero. Since b may be equal to 1, every integer is a rational number...

s with the same distance also form a metric space, but are not complete.
• The positive real numbers with distance function is a complete metric space.
• Any normed vector space
Normed vector space
In mathematics, with 2- or 3-dimensional vectors with real-valued entries, the idea of the "length" of a vector is intuitive and can easily be extended to any real vector space Rn. The following properties of "vector length" are crucial....

is a metric space by defining , see also relation of norms and metrics. (If such a space is complete, we call it a Banach space
Banach space
In mathematics, Banach spaces is the name for complete normed vector spaces, one of the central objects of study in functional analysis. A complete normed vector space is a vector space V with a norm ||·|| such that every Cauchy sequence in V has a limit in V In mathematics, Banach spaces is the...

.) Examples:
• The Manhattan norm gives rise to the Manhattan distance, where the distance between any two points, or vectors, is the sum of the differences between corresponding coordinates.
• The maximum norm gives rise to the Chebyshev distance
Chebyshev distance
In mathematics, Chebyshev 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...

or chessboard distance, the minimal number of moves a chess king
King (chess)
In chess, the king is the most important piece. The object of the game is to trap the opponent's king so that its escape is not possible . If a player's king is threatened with capture, it is said to be in check, and the player must remove the threat of capture on the next move. If this cannot be...

would take to travel from to .
• The British Rail
British Rail
British Railways , which from 1965 traded as British Rail, was the operator of most of the rail transport in Great Britain between 1948 and 1997. It was formed from the nationalisation of the "Big Four" British railway companies and lasted until the gradual privatisation of British Rail, in stages...

metric (also called the Post Office
Post office
A post office is a facility forming part of a postal system for the posting, receipt, sorting, handling, transmission or delivery of mail.Post offices offer mail-related services such as post office boxes, postage and packaging supplies...

metric or the SNCF
SNCF
The SNCF , is France's national state-owned railway company. SNCF operates the country's national rail services, including the TGV, France's high-speed rail network...

metric) on a normed vector space
Normed vector space
In mathematics, with 2- or 3-dimensional vectors with real-valued entries, the idea of the "length" of a vector is intuitive and can easily be extended to any real vector space Rn. The following properties of "vector length" are crucial....

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 (or letters) to proceed via London
London
London is the capital city of :England and the :United Kingdom, the largest metropolitan area in the United Kingdom, and the largest urban zone in the European Union by most measures. Located on the River Thames, London has been a major settlement for two millennia, its history going back to its...

(or Paris
Paris
Paris is the capital and largest city in France, situated on the river Seine, in northern France, at the heart of the Île-de-France region...

) irrespective of their final destination.
• If is a metric space and is a subset
Subset
In mathematics, especially in set theory, a set A is a subset of a set B if A is "contained" inside B. A and B may coincide. The relationship of one set being a subset of another is called inclusion or sometimes containment...

of , then becomes a metric space by restricting the domain of to .
• The discrete metric, where if and otherwise, is a simple but important example, and can be applied to all non-empty sets. This, in particular, shows that for any non-empty set, there is always a metric space associated to it. Using this metric, any point is an open ball, and therefore every subset is open and the space has the discrete topology.
• A finite metric space is a metric space having a finite number of points. Not every finite metric space can be isometrically
Isometric
The term isometric comes from the Greek for "having equal measurement".isometric may mean:* Isometric projection , a method for the visual representation of three-dimensional objects in two dimensions; a form of orthographic projection, or more specifically, an axonometric projection.* Isometry and...

embedded in a Euclidean space
Euclidean space
In mathematics, Euclidean space is the Euclidean plane and three-dimensional space of Euclidean geometry, as well as the generalizations of these notions to higher dimensions...

.
• The hyperbolic plane is a metric space. More generally:
• If is any connected
Connectedness
In mathematics, connectedness is used to refer to various properties meaning, in some sense, "all one piece". When a mathematical object has such a property, we say it is connected; otherwise it is disconnected...

Riemannian manifold
Riemannian manifold
In Riemannian geometry and the differential geometry of surfaces, a Riemannian manifold or Riemannian space is a real differentiable manifold M in which each tangent space is equipped with an inner product g, a Riemannian metric, which varies smoothly from point to point...

, then we can turn into a metric space by defining the distance of two points as the infimum
Infimum
In mathematics, the infimum of a subset S of some partially ordered set T is the greatest element of T that is less than or equal to all elements of S. Consequently the term greatest lower bound is also commonly used...

of the lengths of the paths (continuously differentiable curve
Curve
In mathematics, a curve is, generally speaking, an object similar to a line but which is not required to be straight...

s) connecting them.
• If is some set and is a metric space, then, the set of all bounded function
Bounded function
In mathematics, a function f defined on some set X with real or complex values is called bounded, if the set of its values is bounded. In other words, there exists a real number M...

s (i.e. those functions whose image is a bounded subset of ) can be turned into a metric space by defining for any two bounded functions and (where is supremum
Supremum
In mathematics, given a subset S of a totally or partially ordered set T, the supremum of S, if it exists, is the least element of T that is greater than or equal to every element of S. Consequently, the supremum is also referred to as the least upper bound . If the supremum exists, it is unique...

. This metric is called the uniform metric or supremum metric, and If is complete, then this function space
Function space
In mathematics, a function space is a set of functions of a given kind from a set X to a set Y. It is called a space because in many applications it is a topological space, a vector space, or both.-Examples:...

is complete as well. If X is also a topological space, then set of all bounded continuous functions from to (endowed with the uniform metric), will also be a complete metric if M is.
• If is an undirected connected graph
Graph theory
In mathematics and computer science, graph theory is the study of graphs, mathematical structures used to model pairwise relations between objects from a certain collection. A "graph" in this context refers to a collection of vertices or 'nodes' and a collection of edges that connect pairs of...

, then the set of vertices of can be turned into a metric space by defining to be the length of the shortest path connecting the vertices and . In geometric group theory
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 groups and topological and geometric properties of spaces on which these groups act .Another important...

this is applied to the Cayley graph
Cayley graph
In mathematics, a Cayley graph, also known as a Cayley colour graph, Cayley diagram, group diagram, or colour group is a graph that encodes the abstract structure of a group. Its definition is suggested by Cayley's theorem and uses a specified, usually finite, set of generators for the group...

of a group, yielding the word metric.
• The Levenshtein distance
Levenshtein distance
In information theory and computer science, the Levenshtein distance is a string metric for measuring the amount of difference between two sequences...

is a measure of the dissimilarity between two strings
String (computer science)
In formal languages, which are used in mathematical logic and theoretical computer science, a string is a finite sequence of symbols that are chosen from a set or alphabet....

and , defined as the minimal number of character deletions, insertions, or substitutions required to transform into . 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
Edit distance
In information theory and computer science, the edit distance between two strings of characters generally refers to the Levenshtein distance. However, according to Nico Jacobs, “The term ‘edit distance’ is sometimes used to refer to the distance in which insertions and deletions have equal cost and...

.
• Given a metric space and an increasing concave function
Concave function
In mathematics, a concave function is the negative of a convex function. A concave function is also synonymously called concave downwards, concave down, convex upwards, convex cap or upper convex.-Definition:...

such that if and only if , then is also a metric on .
• Given an injective function
Injective function
In mathematics, an injective function is a function that preserves distinctness: it never maps distinct elements of its domain to the same element of its codomain. In other words, every element of the function's codomain is mapped to by at most one element of its domain...

from any set to a metric space , defines a metric on .
• Using T-theory
T-theory
T-theory is a branch of discrete mathematics dealing with analysis of trees and discrete metric spaces.-General history:As per Andreas Dress, T-theory originated from a question raised by Manfred Eigen, a recipient of the Nobel Prize in Chemistry, in the late seventies. He was trying to fit twenty...

, the tight span
Tight span
In metric geometry, the metric envelope or tight span of a metric space M is an injective metric space into which M can be embedded. In some sense it consists of all points "between" the points of M, analogous to the convex hull of a point set in a Euclidean space. The tight span is also sometimes...

of a metric space is also a metric space. The tight span is useful in several types of analysis.
• The set of all by matrices
Matrix (mathematics)
In mathematics, a matrix is a rectangular array of numbers, symbols, or expressions. The individual items in a matrix are called its elements or entries. An example of a matrix with six elements isMatrices of the same size can be added or subtracted element by element...

over some field
Field (mathematics)
In abstract algebra, a field is a commutative ring whose nonzero elements form a group under multiplication. As such it is an algebraic structure with notions of addition, subtraction, multiplication, and division, satisfying certain axioms...

is a metric space with respect to the rank
Rank (linear algebra)
The column rank of a matrix A is the maximum number of linearly independent column vectors of A. The row rank of a matrix A is the maximum number of linearly independent row vectors of A...

distance .
• The Helly metric  is used in game theory
Game theory
Game theory is a mathematical method for analyzing calculated circumstances, such as in games, where a person’s success is based upon the choices of others...

.

## Open and closed sets, topology and convergence

Every metric space is a topological space
Topological space
Topological spaces are mathematical structures that allow the formal definition of concepts such as convergence, connectedness, and continuity. They appear in virtually every branch of modern mathematics and are a central unifying notion...

in a natural manner, and therefore all definitions and theorems about general topological spaces also apply to all metric spaces.

About any point in a metric space we define the open ball of radius about as the set
These open balls generate a topology on M, making it a topological space
Topological space
Topological spaces are mathematical structures that allow the formal definition of concepts such as convergence, connectedness, and continuity. They appear in virtually every branch of modern mathematics and are a central unifying notion...

.

Explicitly, a subset of is called open
Open set
The concept of an open set is fundamental to many areas of mathematics, especially point-set topology and metric topology. Intuitively speaking, a set U is open if any point x in U can be "moved" a small amount in any direction and still be in the set U...

if for every in there exists an such that is contained in . The complement
Complement (set theory)
In set theory, a complement of a set A refers to things not in , A. The relative complement of A with respect to a set B, is the set of elements in B but not in A...

of an open set is called closed
Closed set
In geometry, topology, and related branches of mathematics, a closed set is a set whose complement is an open set. In a topological space, a closed set can be defined as a set which contains all its limit points...

. A neighborhood of the point is any subset of that contains an open ball about as a subset.

A topological space which can arise in this way from a metric space is called a metrizable space; see the article on metrization theorems for further details.

A sequence
Sequence
In mathematics, a sequence is an ordered list of objects . Like a set, it contains members , and the number of terms is called the length of the sequence. Unlike a set, order matters, and exactly the same elements can appear multiple times at different positions in the sequence...

() in a metric space is said to converge
Limit of a sequence
The limit of a sequence is, intuitively, the unique number or point L such that the terms of the sequence become arbitrarily close to L for "large" values of n...

to the limit iff
IFF
IFF, Iff or iff may refer to:Technology/Science:* Identification friend or foe, an electronic radio-based identification system using transponders...

for every , there exists a natural number N such that for all . Equivalently, one can use the general definition of convergence available in all topological spaces.

A subset of the metric space is closed iff every sequence in that converges to a limit in has its limit in .

### Complete spaces

A metric space is said to be complete if every Cauchy sequence
Cauchy sequence
In mathematics, a Cauchy sequence , named after Augustin-Louis Cauchy, is a sequence whose elements become arbitrarily close to each other as the sequence progresses...

converges in . That is to say: if as both and independently go to infinity, then there is some with .

Every Euclidean space
Euclidean space
In mathematics, Euclidean space is the Euclidean plane and three-dimensional space of Euclidean geometry, as well as the generalizations of these notions to higher dimensions...

is complete, as is every closed subset of a complete space. The rational numbers, using the absolute value metric , are not complete.

Every metric space has a unique (up to isometry
Isometry
In mathematics, an isometry is a distance-preserving map between metric spaces. Geometric figures which can be related by an isometry are called congruent.Isometries are often used in constructions where one space is embedded in another space...

) completion, which is a complete space that contains the given space as a dense
Dense set
In topology and related areas of mathematics, a subset A of a topological space X is called dense if any point x in X belongs to A or is a limit point of A...

subset. For example, the real numbers are the completion of the rationals.

If is a complete subset of the metric space , then is closed in . Indeed, a space is complete iff
IFF
IFF, Iff or iff may refer to:Technology/Science:* Identification friend or foe, an electronic radio-based identification system using transponders...

it is closed in any containing metric space.

Every complete metric space is a Baire space
Baire space
In mathematics, a Baire space is a topological space which, intuitively speaking, is very large and has "enough" points for certain limit processes. It is named in honor of René-Louis Baire who introduced the concept.- Motivation :...

.

### Bounded and totally bounded spaces

A metric space M is called bounded if there exists some number r, such that d(x,y) ≤ r for all x and y in M. The smallest possible such r is called the diameter
Diameter
In geometry, a diameter of a circle is any straight line segment that passes through the center of the circle and whose endpoints are on the circle. The diameters are the longest chords of the circle...

of M. The space M is called precompact or totally bounded 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 interval
Interval (mathematics)
In mathematics, a interval is a set of real numbers with the property that any number that lies between two numbers in the set is also included in the set. For example, the set of all numbers satisfying is an interval which contains and , as well as all numbers between them...

s in the space of real number
Real number
In mathematics, a real number is a value that represents a quantity along a continuum, such as -5 , 4/3 , 8.6 , √2 and π...

s and occasionally regions in a Euclidean space Rn 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.

### Compact spaces

A metric space M is compact if every sequence in M has a subsequence
Subsequence
In mathematics, a subsequence is a sequence that can be derived from another sequence by deleting some elements without changing the order of the remaining elements...

that converges to a point in M. This is known as sequential compactness
Sequentially compact space
In mathematics, a topological space is sequentially compact if every sequence has a convergent subsequence. For general topological spaces, the notions of compactness and sequential compactness are not equivalent; they are, however, equivalent for metric spaces....

and, in metric spaces (but not in general topological spaces), is equivalent to the topological notions of countable compactness
Countably compact space
In mathematics a topological space is countably compact if every countable open cover has a finite subcover.-Examples and Properties:A compact space is countably compact...

and compactness
Compact space
In mathematics, specifically general topology and metric topology, a compact space is an abstract mathematical space whose topology has the compactness property, which has many important implications not valid in general spaces...

defined via open covers.

Examples of compact metric spaces include the closed interval [0,1] with the absolute value metric, all metric spaces with finitely many points, and the Cantor set
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 1875 by Henry John Stephen Smith and introduced by German mathematician Georg Cantor in 1883....

. Every closed subset of a compact space is itself compact.

A metric space is compact iff
IFF
IFF, Iff or iff may refer to:Technology/Science:* Identification friend or foe, an electronic radio-based identification system using transponders...

it is complete and totally bounded. This is known as the Heine–Borel theorem
Heine–Borel theorem
In the topology of metric spaces the Heine–Borel theorem, named after Eduard Heine and Émile Borel, states:For a subset S of Euclidean space Rn, the following two statements are equivalent:*S is closed and bounded...

. Note that compactness depends only on the topology, while boundedness depends on the metric.

Lebesgue's number lemma
Lebesgue's number lemma
In topology, Lebesgue's number lemma, named after Henri Lebesgue, is a useful tool in the study of compact metric spaces. It states:Such a number δ is called a Lebesgue number of this cover. The notion of a Lebesgue number itself is useful in other applications as well.-References:-External links:*...

states that for every open cover of a compact metric space M, there exists a "Lebesgue number" δ such that every subset of M of diameter
Diameter
In geometry, a diameter of a circle is any straight line segment that passes through the center of the circle and whose endpoints are on the circle. The diameters are the longest chords of the circle...

< δ is contained in some member of the cover.

Every compact metric space is second countable, and is a continuous image of the Cantor set
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 1875 by Henry John Stephen Smith and introduced by German mathematician Georg Cantor in 1883....

. (The latter result is due to Pavel Alexandrov and Urysohn
Pavel Samuilovich Urysohn
Pavel Samuilovich Urysohn, Pavel Uryson was a Jewish mathematician who is best known for his contributions in the theory of dimension, and for developing Urysohn's Metrization Theorem and Urysohn's Lemma, both of which are fundamental results in topology...

.)

### Locally compact and proper spaces

A metric space is said to be locally compact 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 {y : d(x,y) ≤ r} is compact. Proper spaces are locally compact, but the converse is not true in general.

### Connectedness

A metric space is connected
Connected space
In topology and related branches of mathematics, a connected space is a topological space that cannot be represented as the union of two or more disjoint nonempty open subsets. Connectedness is one of the principal topological properties that is used to distinguish topological spaces...

if the only subsets that are both open and closed are the empty set and itself.

A metric space is path connected
Connectedness
In mathematics, connectedness is used to refer to various properties meaning, in some sense, "all one piece". When a mathematical object has such a property, we say it is connected; otherwise it is disconnected...

if for any two points there exists a continuous map with and .
Every path connected space is connected, but the converse is not true in general.

There are also local versions of these definitions: locally connected space
Locally connected space
In topology and other branches of mathematics, a topological space X islocally connected if every point admits a neighbourhood basis consisting entirely of open, connected sets.-Background:...

s and locally path connected spaces
Connectedness
In mathematics, connectedness is used to refer to various properties meaning, in some sense, "all one piece". When a mathematical object has such a property, we say it is connected; otherwise it is disconnected...

.

Simply connected space
Simply connected space
In topology, a topological space is called simply connected if it is path-connected and every path between two points can be continuously transformed, staying within the space, into any other path while preserving the two endpoints in question .If a space is not simply connected, it is convenient...

s are those that, in a certain sense, do not have "holes".

### Separable spaces

A metric space is separable space if it has a countable dense
Dense set
In topology and related areas of mathematics, a subset A of a topological space X is called dense if any point x in X belongs to A or is a limit point of A...

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 and also to the Lindelöf
Lindelöf space
In mathematics, a Lindelöf space is a topological space in which every open cover has a countable subcover. The Lindelöf property is a weakening of the more commonly used notion of compactness, which requires the existence of a finite subcover....

property.

## Types of maps between metric spaces

Suppose (M1,d1) and (M2,d2) are two metric spaces.

### Continuous maps

The map f:M1M2 is continuous
if it has one (and therefore all) of the following equivalent properties:
General topological continuity: for every open set U in M2, the preimage f -1(U) is open in M1
This is the general definition of continuity in topology.

Sequential continuity: if (xn) is a sequence in M1 that converges to x in M1, then the sequence (f(xn)) converges to f(x) in M2.
This is sequential continuity, due to Eduard Heine
Eduard Heine
Heinrich Eduard Heine was a German mathematician.Heine became known for results on special functions and in real analysis. In particular, he authored an important treatise on spherical harmonics and Legendre functions . He also investigated basic hypergeometric series...

.

ε-δ definition: for every x in M1 and every ε>0 there exists δ>0 such that for all y in M1 we have
This uses the (ε, δ)-definition of limit, and is due to Augustin Louis Cauchy
Augustin Louis Cauchy
Baron Augustin-Louis Cauchy was a French mathematician who was an early pioneer of analysis. He started the project of formulating and proving the theorems of infinitesimal calculus in a rigorous manner, rejecting the heuristic principle of the generality of algebra exploited by earlier authors...

.

Moreover, f is continuous if and only if it is continuous on every compact subset of M1.

The image
Image (mathematics)
In mathematics, an image is the subset of a function's codomain which is the output of the function on a subset of its domain. Precisely, evaluating the function at each element of a subset X of the domain produces a set called the image of X under or through the function...

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 ƒ : M1 → M2 is uniformly continuous
Uniform continuity
In mathematics, a function f is uniformly continuous if, roughly speaking, it is possible to guarantee that f and f be as close to each other as we please by requiring only that x and y are sufficiently close to each other; unlike ordinary continuity, the maximum distance between x and y cannot...

if for every ε > 0 there exists δ > 0 such that

Every uniformly continuous map ƒ : M1 → M2 is continuous. The converse is true if M1 is compact (Heine–Cantor theorem
Heine–Cantor theorem
In mathematics, the Heine–Cantor theorem, named after Eduard Heine and Georg Cantor, states that if M and N are metric spaces and M is compact then every continuous functionis uniformly continuous....

).

Uniformly continuous maps turn Cauchy sequences in M1 into Cauchy sequences in M2. For continuous maps this is generally wrong; for example, a continuous map
from the open interval (0,1) onto the real line turns some Cauchy sequences into unbounded sequences.

### Lipschitz-continuous maps and contractions

Given a number K > 0, the map ƒ : M1 → M2 is K-Lipschitz continuous
Lipschitz continuity
In mathematical analysis, Lipschitz continuity, named after Rudolf Lipschitz, is a strong form of uniform continuity for functions. Intuitively, a Lipschitz continuous function is limited in how fast it can change: for every pair of points on the graph of this function, the absolute value of the...

if

Every Lipschitz-continuous map is uniformly continuous, but the converse is not true in general.

If K < 1, then ƒ is called a contraction
Contraction mapping
In mathematics, a contraction mapping, or contraction, on a metric space is a function f from M to itself, with the property that there is some nonnegative real number k...

. Suppose M2 = M1 and M1 is complete. If ƒ is a contraction, then ƒ admits a unique fixed point (Banach fixed point theorem
Banach fixed point theorem
In mathematics, the Banach fixed-point theorem is an important tool in the theory of metric spaces; it guarantees the existence and uniqueness of fixed points of certain self-maps of metric spaces, and provides a constructive method to find those fixed points...

). If M1 is compact, the condition can be weakened a bit: ƒ admits a unique fixed point if
for all .

### Isometries

The map f:M1M2 is an isometry
Isometry
In mathematics, an isometry is a distance-preserving map between metric spaces. Geometric figures which can be related by an isometry are called congruent.Isometries are often used in constructions where one space is embedded in another space...

if
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 : M1 → M2 is a quasi-isometry
Quasi-isometry
In mathematics, a quasi-isometry is a means to compare the large-scale structure of metric spaces. The concept is especially important in Gromov's geometric group theory.-Definition:...

if there exist constants A ≥ 1 and B ≥ 0 such that

and a constant C ≥ 0 such that every point in M2 has a distance at most C from some point in the image f(M1).

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
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 groups and topological and geometric properties of spaces on which these groups act .Another important...

in relation to the word metric.

## Notions of metric space equivalence

Given two metric spaces (M1, d1) and (M2, d2):
• They are called homeomorphic (topologically isomorphic) if there exists a homeomorphism
Homeomorphism
In the mathematical field of topology, a homeomorphism or topological isomorphism or bicontinuous function is a continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isomorphisms in the category of topological spaces—that is, they are...

between them (i.e., a bijection
Bijection
A bijection is a function giving an exact pairing of the elements of two sets. A bijection from the set X to the set Y has an inverse function from Y to X. If X and Y are finite sets, then the existence of a bijection means they have the same number of elements...

continuous in both directions).

• They are called uniformic (uniformly isomorphic) if there exists a uniform isomorphism
Uniform isomorphism
In the mathematical field of topology a uniform isomorphism or uniform homeomorphism is a special isomorphism between uniform spaces which respects uniform properties.-Definition:...

between them (i.e., a bijection
Bijection
A bijection is a function giving an exact pairing of the elements of two sets. A bijection from the set X to the set Y has an inverse function from Y to X. If X and Y are finite sets, then the existence of a bijection means they have the same number of elements...

uniformly continuous in both directions).

• They are called isometric if there exists a bijective isometry 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 space
Hausdorff space
In topology and related branches of mathematics, a Hausdorff space, separated space or T2 space is a topological space in which distinct points have disjoint neighbourhoods. Of the many separation axioms that can be imposed on a topological space, the "Hausdorff condition" is the most frequently...

s and hence normal
Normal space
In topology and related branches of mathematics, a normal space is a topological space X that satisfies Axiom T4: every two disjoint closed sets of X have disjoint open neighborhoods. A normal Hausdorff space is also called a T4 space...

(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-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 between the point and the set. If (M,d) is a metric space, S is a subset
Subset
In mathematics, especially in set theory, a set A is a subset of a set B if A is "contained" inside B. A and B may coincide. The relationship of one set being a subset of another is called inclusion or sometimes containment...

of M and x is a point of M, we define the distance from x to S as
d(x,S) = inf
Infimum
In mathematics, the infimum of a subset S of some partially ordered set T is the greatest element of T that is less than or equal to all elements of S. Consequently the term greatest lower bound is also commonly used...

{d(x,s) : sS}

Then d(x, S) = 0 if and only if x belongs to the closure
Closure (topology)
In mathematics, the closure of a subset S in a topological space consists of all points in S plus the limit points of S. Intuitively, these are all the points that are "near" S. A point which is in the closure of S is a point of closure of S...

of S. Furthermore, we have the following generalization of the triangle inequality:
d(x,S) ≤ d(x,y) + d(y,S)

which in particular shows that the map is continuous.

Given two subsets S and T of M, we define their Hausdorff distance
Hausdorff distance
In 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 compact subsets of a metric space into a metric space in its own right...

to be
dH(S,T) = max { sup
Supremum
In mathematics, given a subset S of a totally or partially ordered set T, the supremum of S, if it exists, is the least element of T that is greater than or equal to every element of S. Consequently, the supremum is also referred to as the least upper bound . If the supremum exists, it is unique...

{d(s,T) : sS}, sup {d(t,S) : tT} }

In general, the Hausdorff distance dH(S,T) 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 dH turns the set K(M) of all non-empty compact subsets of M into a metric space. One can show that K(M) is complete if M is complete.
(A different notion of convergence of compact subsets is given by the Kuratowski convergence
Kuratowski convergence
In mathematics, Kuratowski convergence is a notion of convergence for sequences of compact subsets of metric spaces, named after the Polish mathematician Kazimierz Kuratowski. Intuitively, the Kuratowski limit of a sequence of sets is where the sets "accumulate".-Definitions:Let be a metric space...

.)

One can then define the 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 set of all (isometry classes of) compact metric spaces becomes a metric space in its own right.

## Product metric spaces

If are metric spaces, and N is the Euclidean norm on Rn, then is a metric space, where the product metric is defined by

and the induced topology agrees with the product topology
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...

. By the equivalence of norms in finite dimensions, an equivalent metric is obtained if N is the taxicab norm, a p-norm, the max 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

An uncountable product of metric spaces need not be metrizable. For example, is not first-countable
First-countable space
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...

and thus isn't metrizable.

### Continuity of distance

It is worth noting that in the case of a single space , the distance map (from the definition) is uniformly continuous with respect to any of the above product metrics , and in particular is continuous with respect to the product topology of .

## Quotient metric spaces

If M is a metric space with metric d, and ~ is an equivalence relation
Equivalence relation
In mathematics, an equivalence relation is a relation that, loosely speaking, partitions a set so that every element of the set is a member of one and only one cell of the partition. Two elements of the set are considered equivalent if and only if they are elements of the same cell...

on M, then we can endow the quotient set M/~ with the following (pseudo)metric. Given two equivalence classes [x] and [y], we define

where the infimum
Infimum
In mathematics, the infimum of a subset S of some partially ordered set T is the greatest element of T that is less than or equal to all elements of S. Consequently the term greatest lower bound is also commonly used...

is taken over all finite sequences and with , , . In general this will only define a pseudometric
Pseudometric space
In mathematics, a pseudometric space is a generalized metric space in which the distance between two distinct points can be zero. In the same way as every normed space is a metric space, every seminormed space is a pseudometric space...

, i.e. does not necessarily imply that . However for nice equivalence relations (e.g., those given by gluing together polyhedra along faces), it is a metric. Moreover if M is a compact space
Compact space
In mathematics, specifically general topology and metric topology, a compact space is an abstract mathematical space whose topology has the compactness property, which has many important implications not valid in general spaces...

, then the induced topology on M/~ is the quotient topology
Quotient space
In topology and related areas of mathematics, a quotient space is, intuitively speaking, the result of identifying or "gluing together" certain points of a given space. The points to be identified are specified by an equivalence relation...

.

The quotient metric d is characterized by the following universal property
Universal property
In various branches of mathematics, a useful construction is often viewed as the “most efficient solution” to a certain problem. The definition of a universal property uses the language of category theory to make this notion precise and to study it abstractly.This article gives a general treatment...

. If is a metric map between metric spaces (that is, for all x, y) satisfying f(x)=f(y) whenever then the induced function , given by , is a metric map

A topological space is sequential
Sequential space
In topology and related fields of mathematics, a sequential space is a topological space that satisfies a very weak axiom of countability. Sequential spaces are the most general class of spaces for which sequences suffice to determine the topology....

if and only if it is a quotient of a metric space.

## Generalizations of metric spaces

• Every metric space is a uniform space
Uniform space
In the mathematical field of topology, a uniform space is a set with a uniform structure. Uniform spaces are topological spaces with additional structure which is used to define uniform properties such as completeness, uniform continuity and uniform convergence.The conceptual difference between...

in a natural manner, and every uniform space is naturally a topological space
Topological space
Topological spaces are mathematical structures that allow the formal definition of concepts such as convergence, connectedness, and continuity. They appear in virtually every branch of modern mathematics and are a central unifying notion...

. Uniform and topological spaces can therefore be regarded as generalizations of metric spaces.

• If we consider the first definition of a metric space given above and relax the second requirement, or remove the third or fourth, we arrive at the concepts of a pseudometric space
Pseudometric space
In mathematics, a pseudometric space is a generalized metric space in which the distance between two distinct points can be zero. In the same way as every normed space is a metric space, every seminormed space is a pseudometric space...

, a quasimetric space, or a semimetric space.

• If the distance function takes values in the extended real number line
Extended real number line
In mathematics, the affinely extended real number system is obtained from the real number system R by adding two elements: +∞ and −∞ . The projective extended real number system adds a single object, ∞ and makes no distinction between "positive" or "negative" infinity...

R∪{+∞}, but otherwise satisfies all four conditions, then it is called an extended metric and the corresponding space is called an -metric space.

• Approach space
Approach space
In topology, approach spaces are a generalization of metric spaces, based on point-to-set distances, instead of point-to-point distances. They were introduced by in 1989.-Definition:...

s 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
Domain theory
Domain theory is a branch of mathematics that studies special kinds of partially ordered sets commonly called domains. Consequently, domain theory can be considered as a branch of order theory. The field has major applications in computer science, where it is used to specify denotational...

s.

### Metric spaces as enriched categories

The ordered set can be seen as a category by requesting exactly one morphism if and none otherwise. By using as the tensor product and as the identity, it becomes a monoidal category .
Every metric space can now be viewed as a category enriched over :
• Set
• For each set
• The composition morphism will be the unique morphism in given from the triangle inequality
• The identity morphism will be the unique morphism given from the fact that .
• Since is a strict monoidal category, all diagrams that are required for an enriched category commute automatically.

• Aleksandrov–Rassias problem
Aleksandrov–Rassias problem
The theory of isometries in the framework of Banach spaces has its beginning in a paper by Stanisław Mazur and Stanisław M. Ulam in 1932. They proved that each isometry of a normed real linear space onto a normed real linear space is a linear mapping up to translation...

• Basic introduction to the mathematics of curved spacetime
Basic introduction to the mathematics of curved spacetime
The mathematics of general relativity are very complex. In Newton's theories of motions, an object's mass and length remain constant as it changes speed, and the rate of passage of time also remains unchanged. As a result, many problems in Newtonian mechanics can be solved with algebra alone...

• Category of metric spaces
Category of metric spaces
In category-theoretic mathematics, Met is a category that has metric spaces as its objects and metric maps as its morphisms. This is a category because the composition of two metric maps is again a metric map...

• Classical Wiener space
Classical Wiener space
In mathematics, classical Wiener space is the collection of all continuous functions on a given domain , taking values in a metric space . Classical Wiener space is useful in the study of stochastic processes whose sample paths are continuous functions...

• Glossary of Riemannian and metric geometry
Glossary of Riemannian and metric geometry
This is a glossary of some terms used in Riemannian geometry and metric geometry — it doesn't cover the terminology of differential topology....

• Isometry
Isometry
In mathematics, an isometry is a distance-preserving map between metric spaces. Geometric figures which can be related by an isometry are called congruent.Isometries are often used in constructions where one space is embedded in another space...

, contraction mapping
Contraction mapping
In mathematics, a contraction mapping, or contraction, on a metric space is a function f from M to itself, with the property that there is some nonnegative real number k...

and metric map
• Lipschitz continuity
Lipschitz continuity
In mathematical analysis, Lipschitz continuity, named after Rudolf Lipschitz, is a strong form of uniform continuity for functions. Intuitively, a Lipschitz continuous function is limited in how fast it can change: for every pair of points on the graph of this function, the absolute value of the...

• Measure (mathematics)
Measure (mathematics)
In mathematical analysis, a measure on a set is a systematic way to assign to each suitable subset a number, intuitively interpreted as the size of the subset. In this sense, a measure is a generalization of the concepts of length, area, and volume...

• Norm (mathematics)
Norm (mathematics)
In linear algebra, functional analysis and related areas of mathematics, a norm is a function that assigns a strictly positive length or size to all vectors in a vector space, other than the zero vector...

• Premetric space
• Space (mathematics)
• Topology
Topology
Topology is a major area of mathematics concerned with properties that are preserved under continuous deformations of objects, such as deformations that involve stretching, but no tearing or gluing...

• Triangle inequality
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 ....