Metric spaces.html

In mathematics, a metric space is a set where a notion of distance (called 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. 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 connecting them.

The geometric properties 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 also induces topological properties like open and closed sets which leads to the study of even more abstract topological spaces.

1 History
2 Definition
- 2.1 Metric spaces as topological spaces
3 Types of metric spaces
4 Topological properties
5 Examples of metric spaces
6 Notions of metric space equivalence
7 Boundedness and compactness
8 Separation properties and extension of continuous functions
- 8.1 Distance between points and sets
9 Product metric spaces
- 9.1 Continuity of distance
10 Quotient metric spaces
11 See also
12 Notes
13 Sources
14 External links

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 (M,d) where M is a set and d is a metric on M, that is, a function

$d : M \times M \rightarrow \mathbb{R}$

such that for any x, y and z in M

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 M for a metric space if it is clear from the context what metric is used. Relaxing the second requirement, or removing the third or fourth, leads to the concepts of a pseudometric space, a quasimetric space, or a semimetric space. If the function takes values in the extended real number line, but otherwise satisfies above conditions, then it is called an extended metric; the corresponding space is then called an $\infty$ -metric space.

The first of these four conditions actually follows from the other three, since:

2d(x, y) = d(x, y) + d(y, x) ≥ d(x,x) = 0.

Some authors require the set M to be non-empty.

Metric spaces as topological spaces

The treatment of a metric space as a topological space is so consistent that it is almost a part of the definition.

About any point x in a metric space M we define the open ball of radius r (>0) about x as the set

B(x; r) = {y in M : d(x,y) < r}.

These open balls generate a topology on M, making it a topological space. Explicitly, a subset of M is called open if it is a union of (finitely or infinitely many) open balls. (The null set is an open set, and may be thought of as the union of zero balls.) The complement of an open set is called closed. The metric topology on 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. 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. This definition is equivalent to the usual epsilon-delta definition of continuity (which does not refer to the topology), and can also be directly defined using limits of sequences.

Types of metric spaces

A metric space X is said to be complete if every Cauchy sequence converges in X. That is to say that: , as both n and m independently go to infinity, implies that there is some y with .
- A complete metric space is a Baire space.
A metric space $X$ is said to be proper if all closed balls in $X$ are compact.

Topological properties

Every metric space is:

Other topological properties become equivalent in metric spaces. In particular,

second countability, separability and the Lindelöf property are all equivalent.
compactness, sequential compactness, and countable compactness are all equivalent.

Every compact metric space is second countable,¹ and is a continuous image of a Cantor set. (The latter result is due to Alexandroff and Urysohn.)

Examples of metric spaces

The discrete metric, where d(x,y)=0 if x = y and d(x,y)=1 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.
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. The rational numbers with the same distance function are also a metric space, but are not complete.
The hyperbolic plane is a metric space.
Any normed vector space is a metric space by defining d(x, y) = ||y − x||, see also relation of norms and metricshttp://en.wikipedia.org/en/Metric_(mathematics)#Relation_of_norms_and_metrics. (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 maximum norm gives rise to the Chebyshev distance or chessboard distance, the minimal number of moves a chess king would take to travel from x to y.
The British Rail metric (also called the Post Office metric or the SNCF metric) on a normed vector space, given by d(x, y) = ||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(x, y)=f(x)+f(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 (or Paris) irrespective of their final destination.
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.
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.
The Levenshtein distance, also called character edit distance, is a measure of the dissimilarity between two strings u and v. The distance is the minimal number of character deletions, insertions, or substitutions required to transform u into v. This can be thought as a special case of shortest path metric in a graph.
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 (Euclidean metric), √5 (surface path), and 3 (edges path), 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 an increasing concave function f:[0,∞)→[0,∞) such that f(x)=0 if and only if x=0, then f o d is also a metric on X.
Given an injective function f from any set A to a metric space (X,d), d(f(x), f(y)) defines a metric on A.
Using T-theory, the tight span of a metric space is also a metric space. The tight span is useful in several types of analysis.
The set of all n by m matrices over a finite field is a metric space with respect to the rank distance d(X,Y) = rank(Y-X).
The Helly metric is used in game theory.

Notions of metric space equivalence

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 homeomorphic (topologically isomorphic) if there exists a homeomorphism between them (i.e., a bijection continuous in both directions).

They are called uniformic (uniformly isomorphic) if there exists a uniform isomorphism between them (i.e., a bijection uniformly continuous in both directions)

They are called 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 isometric 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 (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.

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 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 (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. This is known as Heine–Borel theorem. 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.

In a metric space, sequential compactness, countable compactness and compactness are all equivalent.

By restricting the metric, any subset of a metric space is a metric space itself (a subspace) with a topology restricted to that set. We call such a subset complete, bounded, totally bounded or compact if it, considered as a metric space, has the corresponding property. Every closed subspace of a complete metric space is complete, and every complete subspace of a metric space is closed. A closed subspace of a metric space, however, need not be complete.

Separation properties and extension of continuous functions

Metric spaces are paracompact² Hausdorff spaces³ 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.

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 $x\mapsto d(x,S)$ is continuous.

Product metric spaces

If $(M_1,d_1),\ldots,(M_n,d_n)$ are metric spaces, and N is the Euclidean norm on Rⁿ, then $\Big(M_1\times \ldots \times M_n, N(d_1,\ldots,d_n)\Big)$ is a metric space, where the product metric is defined by

$N(d_1,...,d_n)\Big((x_1,\ldots,x_n),(y_1,\ldots,y_n)\Big) = N\Big(d_1(x_1,y_1),\ldots,d_n(x_n,y_n)\Big)$ ,

and the induced topology agrees with the product topology. By the equivalence of norms in finite dimensions, an equivalent metric is obtained if N is the 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

$d(x,y)=\sum_{i=1}^\infty \frac1{2^i}\frac{d_i(x_i,y_i)}{1+d_i(x_i,y_i)}$ .

An uncountable product of metric spaces is, in general, not metrizable. For example, $\mathbf{R}^\mathbf{R}$ is not first-countable space (thus cannot be metrizable).

Continuity of distance

It is worth noting that in the case of a single space $(M, d)$ , the distance map $d:M\times M \rightarrow R^+$ (from the definition) is uniformly continuous with respect to any of the above product metrics $N (d, d)$ (and in particular, continuous with respect to the product topology of $M\times M$ ).

Quotient metric spaces

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

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

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 $p 1 = x$ , $q n = y$ , $[q_i]=[p_{i+1}], i=1,2,\dots n-1$ . 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.

The quotient metric d is characterized by the following universal property. If $f:(M,d)\longrightarrow(X,\delta)$ is a metric 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 $x\sim y,$ then the induced function $\overline{f}:M/\sim\longrightarrow X$ , given by $\overline{f}([x])=f(x)$ , is a metric map $\overline{f}:(M/\sim,d')\longrightarrow (X,\delta).$

A topological space is sequential if and only if it is a quotient of a metric space.⁴

Notes

^ PlanetMath: a compact metric space is second countable
^ Rudin, Mary Ellen. A new proof that metric spaces are paracompact. Proceedings of the American Mathematical Society, Vol. 20, No. 2. (Feb., 1969), p. 603.
^ metric spaces are Hausdorff on PlanetMath
^ Goreham, Anthony. Sequential convergence in Topological Spaces. Honours' Dissertation, Queen's College, Oxford (April, 2001), p. 14

Sources

Victor Bryant, Metric Spaces: Iteration and Application, Cambridge University Press, 1985, ISBN 0-521-31897-1.
Dmitri Burago, Yu D Burago, Sergei Ivanov, A Course in Metric Geometry, American Mathematical Society, 2001, ISBN 0-8218-2129-6.
Mícheál Ó Searcóid, Metric Spaces, Springer Undergraduate Mathematics Series, 2006, ISBN 1-84628-369-8.
Eric W. Weisstein, Product Metric at MathWorld.

External links

Far and near — several examples of distance functions at cut-the-knot

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