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Encyclopedia > Metric space

In mathematics, a metric space is a set where a notion of distance 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. The Euclidean metric of this space defines the distance between two points as the length of the straight line connecting them. The geometry of the space depends 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. Euclid, a famous Greek mathematician known as the father of geometry, is shown here in detail from The School of Athens by Raphael. ... In mathematics, a set can be thought of as any collection of distinct things considered as a whole. ... The distance between two points is the length of a straight line segment between them. ... In mathematics, Euclidean space is a generalization of the 2- and 3-dimensional spaces studied by Euclid. ... The Euclidean distance of two points x = (x1,...,xn) and y = (y1,...,yn) in Euclidean n-space is computed as It is the ordinary distance between the two points that one would measure with a ruler, which can be proven by repeated application of the Pythagorean theorem. ... A line, or straight line, is, roughly speaking, an (infinitely) thin, (infinitely) long, straight geometrical object, i. ... Table of Geometry, from the 1728 Cyclopaedia. ... The term non-Euclidean geometry (also spelled: non-Euclidian geometry) describes both hyperbolic and elliptic geometry, which are contrasted with Euclidean geometry. ... For a non-technical introduction to the topic, please see Introduction to General relativity. ...


A metric space induces topological properties like open and closed sets which leads to the study of even more abstract topological spaces. In topology and related areas of mathematics a topological property or topological invariant is a property of a topological space which is invariant under homeomorphisms. ... In topology and related fields of mathematics, a set U is called open if, intuitively speaking, you can wiggle or change any point x in U by a small amount in any direction and still be inside U. In other words, if x is surrounded only by elements of U... In topology and related branches of mathematics, a closed set is a set whose complement is open. ... Topological spaces are structures that allow one to formalize concepts such as convergence, connectedness and continuity. ...

Contents


History

Maurice Fréchet introduced metric spaces in his work Sur quelques points du calcul fonctionnel, Rendic. Circ. Mat. Palermo 22(1906) 1-74. Maurice Fréchet (born September 2, 1878, died June 4, 1973) was a French mathematician. ...


Definition

A metric space is a 2-tuple (X,d) where X is a set and d is a metric on X, that is, a function In mathematics, a tuple is a finite sequence of objects (a list of a limited number of objects). ... In mathematics, a set can be thought of as any collection of distinct things considered as a whole. ... In mathematics a metric or distance is a function which assigns a distance to elements of a set. ...

d : X × XR

such that

  1. d(x, y) ≥ 0     (non-negativity)
  2. d(x, y) = 0   if and only if   x = y     (identity of indiscernibles)
  3. d(x, y) = d(y, x)     (symmetry)
  4. d(x, z) ≤ d(x, y) + d(y, z)     (triangle inequality).

The function d is also called distance function or simply distance. Often d is omitted and one just writes X for a metric space if it is clear from the context what metric is used. If we relax the second condition to allow zero distance between two distinct points, then the space is known as semi-metric space or pseudo-metric space. In mathematics, triangle inequality is the theorem stating that for any triangle, the measure of a given side must be less than the sum of the other two sides but greater than the difference between the two sides. ...


Examples

  • The real numbers with the distance function d(x, y) = |yx| given by the absolute value, and more generally Euclidean n-space with the Euclidean distance, are complete metric spaces.
  • Hyperbolic space.
  • Any normed vector space is a metric space by defining d(x, y) = ||yx||, see also distances based on norms. (If such a space is complete, we call it a Banach space). Example:
    • the Manhattan norm gives rise to the Manhattan distance, where the distance between any two points, or vectors, is the sum of the distances between corresponding coordinates.
  • The discrete metric, where d(x,y)=1 for all x not equal to y and d(x,y)=0 otherwise, is a simple but important example, and can be applied to all sets.
  • The British Rail metric (also called the Post Office metric or the SNCF metric) on a normed vector space, given by d(xy)=||x|| + ||y|| for distinct points x and y, and d(x, x) = 0. More generally ||.|| can be replaced with a function f taking an arbitrary set S to non-negative reals and taking the value 0 at most once: then the metric is defined on S by d(xy)=f(x)+f(y). The name alludes to the tendency of railway journeys (or letters) to proceed via London (or Paris) irrespective of their final destination.
  • The Chessboard distance, the number of moves a chess king would take to travel from x to y.
  • If X is some set and M is a metric space, then the set of all bounded functions f : XM (i.e. those functions whose image is a bounded subset of M) can be turned into a metric space by defining d(f, g) = supx in X d(f(x), g(x)) for any bounded functions f and g. If M is complete, then this space is complete as well.
  • The Levenshtein distance, also called character edit distance, is a measure of the similarity between two strings u and v. The distance is the minimal number of deletions, insertions, or substitutions required to transform u into v.
  • If X is a topological (or metric) space and M is a metric space, then the set of all bounded continuous functions from X to M forms a metric space if we define the metric as above: d(f, g) = supx in X d(f(x), g(x)) for any bounded continuous functions f and g. If M is complete, then this space is complete as well.
  • If M is a connected Riemannian manifold, then we can turn M into a metric space by defining the distance of two points as the infimum of the lengths of the paths (continuously differentiable curves) connecting them.
  • If G is an undirected connected graph, then the set V of vertices of G can be turned into a metric space by defining d(x, y) to be the length of the shortest path connecting the vertices x and y.
  • Similarly (apart from mathematical details):
    • For any system of roads and terrains the distance between two locations can be defined as the length of the shortest route. To be a metric there should not be one-way roads. Examples include some mentioned above: the Manhattan norm, the British Rail metric, and the Chessboard distance.
    • More generally, for any system of roads and terrains, with given maximum possible speed at any location, the "distance" between two locations can be defined as the time the fastest route takes. To be a metric there should not be one-way roads, and the maximum speed should not depend on direction. The direction at A to B can be defined, not necessarily uniquely, as the direction of the "shortest" route, i.e., in which the "distance" reduces 1 second per second when travelling at the maximum speed.
  • Similarly, in 3D, the metrics on the surface of a polyhedron include the ordinary metric, and the distance over the surface; a third metric on the edges of a polyhedron is one where the "paths" are the edges. For example, the distance between opposite vertices of a unit cube is √3, √5, and 3, respectively.
  • If M is a metric space, we can turn the set K(M) of all compact subsets of M into a metric space by defining the Hausdorff distance d(X, Y) = inf{r : for every x in X there exists a y in Y with d(x, y) < r and for every y in Y there exists an x in X such that d(x, y) < r)}. In this metric, two elements are close to each other if every element of one set is close to some element of the other set. One can show that K(M) is complete if M is complete.
  • The set of all (isometry classes of) compact metric spaces form a metric space with respect to Gromov-Hausdorff distance.
  • Given a metric space (X,d) and a increasing concave function f:[0,∞)→[0,∞) such that f(x)=0 iff x=0, then f o d is also a metric on X.
  • Given a injective function f from any set A to a metric space (X,d), d(f(x), f(y)) defines a metric on A.

In mathematics, the real numbers are intuitively defined as numbers that are in one-to-one correspondence with the points on an infinite line—the number line. ... In mathematics, the absolute value (or modulus1) of a real number is its numerical value without regard to its sign. ... In mathematics, Euclidean space is a generalization of the 2- and 3-dimensional spaces studied by Euclid. ... In mathematics the Euclidean distance or Euclidean metric is the ordinary distance between the two points that one would measure with a ruler, which can be proven by repeated application of the Pythagorean theorem. ... In mathematical analysis, a metric space M is said to be complete (or Cauchy) if every Cauchy sequence of points in M has a limit that is also in M. Intuitively, a space is complete if it doesnt have any holes, if there arent any points missing. For... In mathematics, hyperbolic n-space, denoted Hn, is the maximally symmetric, simply connected, n-dimensional Riemannian manifold with constant sectional curvature −1. ... Link titleIn mathematics, with 2- or 3-dimensional vectors with real-valued entries, the idea of the length of a vector is intuitive and can be easily extended to any real vector space Rn. ... The distance between two points is the length of a straight line segment between them. ... In mathematics, Banach spaces, named after Stefan Banach who studied them, are one of the central objects of study in functional analysis. ... In linear algebra, functional analysis and related areas of mathematics, a norm is a function which assigns a positive length or size to all vectors in a vector space, other than the zero vector. ... Taxicab geometry, considered by Hermann Minkowski in the 19th century, is a form of geometry in which the usual 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 coordinates. ... Logo of British Rail British Railways (BR), later rebranded as British Rail, ran the British railway system from the nationalisation of the Big Four British railway companies in 1948 until its privatisation in stages between 1994 and 1997. ... Small-town post office and town hall in Lockhart, Alabama A post office is a facility (in most countries, a government one) where the public can purchase postage stamps for mailing correspondence or merchandise, and also drop off or pick up packages or other special-delivery items. ... An SNCF multiple unit. ... Link titleIn mathematics, with 2- or 3-dimensional vectors with real-valued entries, the idea of the length of a vector is intuitive and can be easily extended to any real vector space Rn. ... For other uses, see London (disambiguation). ... The Eiffel Tower, the international symbol of the city For other uses, see Paris (disambiguation). ... Chessboard distance is a metric that can be used to measure distances. ... 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 mathematical analysis and related areas of mathematics, a set is called bounded, if it is, in a certain sense, of finite size. ... In information theory, the Levenshtein distance or edit distance between two strings is given by the minimum number of operations needed to transform one string into the other, where an operation is an insertion, deletion, or substitution of a single character. ... In information theory, the Levenshtein distance or edit distance between two strings is given by the minimum number of operations needed to transform one string into the other, where an operation is an insertion, deletion, or substitution. ... Topological spaces are structures that allow one to formalize concepts such as convergence, connectedness and continuity. ... In topology and related areas of mathematics a continuous function is a morphism between topological spaces. ... In topology and related branches of mathematics, a topological space X is said to be disconnected if it is the union of two disjoint nonempty open sets. ... In Riemannian geometry, a Riemannian manifold is a real differentiable manifold in which each tangent space is equipped with an inner product in a manner which varies smoothly from point to point. ... In mathematics the infimum of a subset of some set is the greatest element, not necessarily in the subset, that is smaller than all other elements of the subset. ... In mathematics, the concept of a curve tries to capture the intuitive idea of a geometrical one-dimensional and continuous object. ... A labeled graph with 6 vertices and 7 edges. ... The shape of each panel of this road sign, and the broken lines at the ends, represents an arrow; a space-consuming central bar of the arrow sign is dispensed with. ... This article is about the geometric shape. ... A unit cube is a 3-dimensional geometric figure that consists of a cube in which all of its dimensions are 1 unit long. ... Hausdorff distance measures how far two compact subsets of a metric space are from each other. ... Gromov-Hausdorff convergence is a notion for convergence of metric spaces which is a generalization of Hausdorff convergence. ...

Metric spaces as topological spaces

In any metric space M we can define the open balls as the sets of the form A synonym for ball (in geometry or topology, and in any dimension) is disk (or disc Geometry In metric geometry, a ball is a set containing all points within a specified distance of a given point. ...

B(x; r) = {y in M : d(x,y) < r},

where x is in M and r is a positive real number, called the radius of the ball. A subset of M which is a union of (finitely or infinitely many) open balls is called an open set. The complement of an open set is called closed. Every metric space is automatically a topological space, the topology being the set of all open sets. 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. In topology and related fields of mathematics, a set U is called open if, intuitively speaking, you can wiggle or change any point x in U by a small amount in any direction and still be inside U. In other words, if x is surrounded only by elements of U... In topology and related branches of mathematics, a closed set is a set whose complement is open. ... Topological spaces are structures that allow one to formalize concepts such as convergence, connectedness and continuity. ... A metrizable space is a topological space that is homeomorphic to a metric space. ...


Since metric spaces are topological spaces, one has a notion of continuous function between metric spaces. Without referring to the topology, this notion can also be directly defined using limits of sequences; this is explained in the article on continuous functions. In topology and related areas of mathematics a continuous function is a morphism between topological spaces. ...


Boundedness and compactness

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 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 equals 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 (the first example above) under which it is bounded and yet not totally bounded. A useful characterisation of compactness for metric spaces is that a metric space is compact if and only if it is complete and totally bounded. Note that compactness depends only on the topology, while boundedness depends on the metric. Diameter is an AAA (authentication, authorization and accounting) protocol for applications such as network access or IP mobility. ... In mathematics, a metric space is a set (or space) where a distance between points is defined. ... In mathematics, a subset of Euclidean space Rn is called compact if it is closed and bounded. ... In mathematics, a subset of Euclidean space Rn is called compact if it is closed and bounded. ...


Note that in the context of intervals in the space of real numbers 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. In mathematics, interval is a concept relating to the sequence and set-membership of one or more numbers. ... In mathematics, the real numbers are intuitively defined as numbers that are in one-to-one correspondence with the points on an infinite line—the number line. ...


By restricting the metric, any subset of a metric space is a metric space itself (a subspace). We call such a subset complete, bounded, totally bounded or compact if it, considered as a metric space, has the corresponding property.


Separation properties and extension of continuous functions

Metric spaces are paracompact Hausdorff spaces and hence normal (indeed they are perfectly normal). An important consequence is that every metric space admits partitions of unity and that every continuous real-valued function defined on a closed subset of a metric space can be extended to a continuous map on the whole space (Tietze extension theorem). It is also true that every real-valued Lipschitz-continuous map defined on a subset of a metric space can be extended to a Lipschitz-continuous map on the whole space. In mathematics, a paracompact space is a topological space in which every open cover admits an open locally finite refinement. ... In topology and related branches of mathematics, a Hausdorff space is a topological space in which points can be separated by neighbourhoods. ... In topology and related branches of mathematics, normal spaces, T4 spaces, and T5 spaces are particularly nice kinds of topological spaces. ... This is a glossary of some terms used in the branch of mathematics known as topology. ... The Tietze extension theorem in topology states that, if X is a normal topological space and f : A → R is a continuous map from a closed subset A of X into the real numbers carrying the standard topology, then there exists a continuous map F : X → R with F(a... In mathematics, a function f : M → N between metric spaces M and N is called Lipschitz continuous (or is said to satisfy a Lipschitz condition) if there exists a constant K > 0 such that d(f(x), f(y)) ≤ K d(x, y) for all x and y in M...


Distance between points and sets

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 of M and x is a point of M, we define the distance from x to S as In topology and related branches of mathematics, Tychonoff spaces and completely regular spaces are particularly nice kinds of topological spaces. ... A is a subset of B, and B is a superset of A. In mathematics, especially in set theory, a set A is a subset of a set B, if A is contained inside B. The relationship of one set being a subset of another is called inclusion. ...

d(x,S) = inf {d(x,s) : sS}

Then d(x, S) = 0 if and only if x belongs to the closure of S. Furthermore, we have the following generalization of the triangle inequality: In mathematics the infimum of a subset of some set is the greatest element, not necessarily in the subset, that is smaller than all other elements of the subset. ... In mathematics, the closure of a set S consists of all points which are intuitively close to S. A point which is in the closure of S is a point of closure of S. The notion of closure is in many ways dual to the notion of interior. ...

d(x,S) ≤ d(x,y) + d(y,S)

which in particular shows that the map xmapsto d(x,S) is continuous.


Equivalence of metric spaces

Comparing two metric spaces one can distinguish various degrees of equivalence. To preserve at least the topological structure induced by the metric, these require at least the existence of a continuous function between them (morphism preserving the topology of the metric spaces). In topology, a continuous function is generally defined as one for which preimages of open sets are open. ... In mathematics, a morphism is an abstraction of a structure-preserving process between two mathematical structures. ...


Given two metric spaces (M1, d1) and (M2, d2):

  • They are called topologically isomorphic (or homeomorphic) if there exists a homeomorphism between them.
  • They are called isometrically isomorphic if there exists a bijective isometry between them. In this case, the two spaces are essentially identical. An isometry is a function f : M1M2 which preserves distances: d2(f(x), f(y)) = d1(x, y) for all x, y in M1. Isometries are necessarily injective.
  • They are called similar if there exists a positive constant k > 0 and a bijective function f, called similarity such that f : M1M2 and d2(f(x), f(y)) = k d1(x, y) for all x, y in M1.
  • They are called similar (of the second type) if there exists a bijective function f, called similarity such that f : M1M2 and d2(f(x), f(y)) = d2(f(u), f(v)) if and only if d1(x, y) = d1(u, v) for all x, y,u, v in M1.

In case of Euclidean space with usual metric the two notions of similarity are equivalent. In the mathematical field of topology a homeomorphism or topological isomorphism (from the Greek words homeos = identical and morphe = shape) is a special isomorphism between topological spaces which respects topological properties. ... In the mathematical field of topology a uniform isomorphism or uniform homeomorphism is a special isomorphism between uniform spaces which respects uniform properties. ... In mathematics, a bijection, bijective function, or one-to-one correspondence is a function that is both injective (one-to-one) and surjective (onto), and therefore bijections are also called one_to_one and onto. ... In mathematics, an isometry, isometric isomorphism or congruence mapping is a distance-preserving isomorphism between metric spaces. ... In mathematics, an injective function (or one-to-one function or injection) is a function which maps distinct input values to distinct output values. ... In mathematics, a bijection, bijective function, or one-to-one correspondence is a function that is both injective (one-to-one) and surjective (onto), and therefore bijections are also called one_to_one and onto. ... Several equivalence relations in mathematics are called similarity. ... In mathematics, a bijection, bijective function, or one-to-one correspondence is a function that is both injective (one-to-one) and surjective (onto), and therefore bijections are also called one_to_one and onto. ... Several equivalence relations in mathematics are called similarity. ... In mathematics, Euclidean space is a generalization of the 2- and 3-dimensional spaces studied by Euclid. ...


Quotient metric space

If M is a metric space with metric d, and ~ is an equivalence relation on M, then we can endow the quotient set M/~ with the following (pseudo)metric. Given two equivalence classes [x] and [y], we define In mathematics, an equivalence relation on a set X is a binary relation on X that is reflexive, symmetric and transitive, i. ...

d'([x],[y]) = inf{d(p_1,q_1)+d(p_2,q_2)+...+d(p_{n},q_{n})}

where the infimum is taken over all finite sequences (p_1, p_2, dots, p_n) and (q_1, q_2, dots, q_n) with [p1] = [x], [qn] = [y], [q_i]=[p_{i+1}], i=1,2,dots n-1. In general this will only define a pseudometric, i.e. d'([x],[y]) = 0 does not necessarily imply that [x]=[y]. 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, then the induced topology on M/~ is the quotient topology. In mathematics the infimum of a subset of some set is the greatest element, not necessarily in the subset, that is smaller than all other elements of the subset. ... In mathematics, a subset of Euclidean space Rn is called compact if it is closed and bounded. ... In topology and related areas of mathematics, a quotient space (also called an identification space) is, intuitively speaking, the result of identifying or gluing together certain points of a given space. ...


The quotient metric d' is characterized by the following universal property. If f:(M,d)longrightarrow(X,delta) is a short map between metric spaces (that is, delta(f(x),f(y))le d(x,y) for all x, y) satisfying f(x)=f(y) whenever xsim y, then the induced function overline{f}:M/simlongrightarrow X, given by overline{f}([x])=f(x), is a short map overline{f}:(M/sim,d')longrightarrow (X,delta). In various branches of mathematics, certain constructions are frequently defined or characterised by an abstract property which requires the existence of a unique morphism under certain conditions. ... In mathematics, a short map is a function f from a metric space X to another metric space Y such that for any we have . Here and denote metrics on and , respectively. ...


See also

This is a glossary of some terms used in Riemannian geometry and metric geometry — it doesnt cover the terminology of differential topology. ... Topology (Greek topos, place and logos, study) is a branch of mathematics concerned with spatial properties preserved under bicontinuous deformation (stretching without tearing or gluing); these are the topological invariants. ... In mathematics, triangle inequality is the theorem stating that for any triangle, the measure of a given side must be less than the sum of the other two sides but greater than the difference between the two sides. ... In mathematics, a function f : D → R defined on a set D of real numbers with real values is called Lipschitz continuous (or is said to satisfy a Lipschitz condition) if there exists a constant K ≥ 0 such that for all in D. The smallest such K is called the... In mathematics, an isometry, isometric isomorphism or congruence mapping is a distance-preserving isomorphism between metric spaces. ... In mathematics, a contraction mapping, or contraction, on a metric space (M,d) is a function f from M to itself, with the property that there is some real number such that, for all x and y in M, The smallest such value of k is called the Lipschitz constant... In mathematics, a short map is a function f from a metric space X to another metric space Y such that for any we have . Here and denote metrics on and , respectively. ... The category Met has metric spaces as objects and short maps as morphisms. ... In linear algebra, functional analysis and related areas of mathematics, a norm is a function which assigns a positive length or size to all vectors in a vector space, other than the zero vector. ...

References

  • Dmitri Burago, Iu D Burago, Sergei Ivanov, A Course in Metric Geometry, American Mathematical Society, 2001, ISBN 0821821296.

External links

  • Far and near — several examples of distance functions at cut-the-knot
  • Metric Space — Metric Spaces on Wolfram's MathWorld

  Results from FactBites:
 
Metric space - Wikipedia, the free encyclopedia (1957 words)
In mathematics, a metric space is a set where a notion of distance between elements of the set is defined.
Similarly, in 3D, the metrics on the surface of a polyhedron include the ordinary metric, and the distance over the surface; a third metric on the edges of a polyhedron is one where the "paths" are the edges.
Metric spaces are paracompact Hausdorff spaces and hence normal (indeed they are perfectly normal).
Complete space - Wikipedia, the free encyclopedia (1204 words)
In mathematical analysis, a metric space M is said to be complete (or Cauchy) if every Cauchy sequence of points in M has a limit that is also in M.
For any metric space M, one can construct a complete metric space M' (which is also denoted as M with a bar over it), which contains M as a dense subspace.
Completely metrizable spaces can be characterized as those spaces which can be written as an intersection of countably many open subsets of some complete metric space.
  More results at FactBites »

 
 

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