Metric space

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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.

A metric space induces topological properties like open and closed sets which leads to the study of even more abstract topological spaces.

1 History
2 Definition
3 Examples
4 Metric spaces as topological spaces
5 Boundedness and compactness
6 Separation properties and extension of continuous functions
- 6.1 Distance between points and sets
7 Equivalence of metric spaces
8 Quotient metric space
9 See also
10 References
11 External links

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History

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

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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

d : X × X → R

such that

d(x, y) ≥ 0 (non-negativity)
d(x, y) = 0 if and only if x = y (identity of indiscernibles)
d(x, y) = d(y, x) (symmetry)
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.

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Examples

The real numbers with the distance function d(x, y) = |y − x| 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) = ||y − x||, 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) on a normed vector space, given by d(x, y)=||x|| + ||y|| for distinct points x and y, and d(x, x) = 0. The name alludes to the tendency of railway journeys (or letters) to proceed via London, which is identified with the origin. However, this comparison breaks down when considering the distance between two stations on the same line.
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 : X → M (i.e. those functions whose image is a bounded subset of M) can be turned into a metric space by defining d(f, g) = sup_{x 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) = sup_{x 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.

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Metric spaces as topological spaces

In any metric space M we can define the open balls as the sets of the form

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.

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.

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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.

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.

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.

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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.

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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

d(x,S) = inf {d(x,s) : s ∈ S}

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:

d(x,S) ≤ d(x,y) + d(y,S)

which in particular shows that the map <math>x\mapsto d(x,S)</math> is continuous.

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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).

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 uniformly isomorphic if there exists a uniform isomorphism 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 : M₁ → M₂ 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 : M₁ → M₂ 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 : M₁ → M₂ 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.

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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

where the infimum is taken over all finite sequences <math>(p_1, p_2, \dots, p_n)</math> and <math>(q_1, q_2, \dots, q_n)</math> with <math>[p_1]=[x]</math>, <math>[q_n]=[y]</math>, <math>[q_i]=[p_{i+1}], i=1,2,\dots n-1</math>. In general this will only define a pseudometric, i.e. <math>d'([x],[y])=0</math> 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.

The quotient metric d' is characterized by the following universal property. If <math>f:(M,d)\longrightarrow(X,\delta)</math> is a short map between metric spaces (that is, <math>\delta(f(x),f(y))\le d(x,y)</math> for all x, y) satisfying f(x)=f(y) whenever <math>x\sim y,</math> then the induced function <math>\overline{f}:M/\sim\longrightarrow X</math>, given by <math>\overline{f}([x])=f(x)</math>, is a short map <math>\overline{f}:(M/\sim,d')\longrightarrow (X,\delta).</math>

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