Metric space
In mathematics, a metric space is a set where a notion of (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.
Contents
- 1 History
- 2 Definition
- 3 Examples of metric spaces
- 4 Open and closed sets, topology and convergence
- 5 Types of metric spaces
- 6 Types of maps between metric spaces
- 7 Notions of metric space equivalence
- 8 Topological properties
- 9 Distance between points and sets; Hausdorff distance and Gromov metric
- 10 Product metric spaces
- 11 Quotient metric spaces
- 12 Generalizations of metric spaces
- 13 See also
- 14 Notes
- 15 References
- 16 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 (which some authors require to be non-empty) and d is a metric on M, that is, a function
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 first condition follows from the other three, since:
- 2d(x, y) = d(x, y) + d(y, x) ≥ d(x,x) = 0.
A more compact but equivalent definition replaces the above four axioms by the following two:
- d(x, y) = 0 if and only if x = y
- d(x, z) ≤ d(x, y) + d(z, y)
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.
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 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.
- Any normed vector space is a metric space by defining d(x, y) = ||y − x||, see also relation of norms and metrics. (If such a space is complete, we call it a Banach space). Examples:
- 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 (M,d) is a metric space and X is a subset of M, then X becomes a metric space by restricting d to X×X.
- 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 hyperbolic plane is a metric space.
- 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. In geometric group theory this is applied to the Cayley graph of a group, yielding the word metric.
- The Levenshtein distance is a measure of the dissimilarity between two strings u and v, defined as the minimal number of character deletions, insertions, or substitutions required to transform u into v. This can be thought as a special case of the shortest path metric in a graph and is one example of an edit 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 m by n matrices over some 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.
Open and closed sets, topology and convergence
Every metric space is a topological space in a natural manner, and therefore all definitions and theorems about topological spaces also apply to all metric spaces.
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 U of M is called open if to every x in U there exists an r>0 such that B(x; r) is contained in U. The complement of an open set is called closed. A neighborhood of the point x is any subset of M that contains an open ball about x 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 (x_{n}) in a metric space M is said to converge to the limit x∈M iff for every r>0 there exists a natural number N such that d(x_{n},x) < r for all n>N. Equivalently, one can use the general definition of convergence available in all topological spaces.
A subset A of the metric space M is closed if and only if every sequence in A that converges to a limit in M has its limit in A.
Types of metric spaces
Complete spaces
A metric space is said to be complete if every Cauchy sequence converges in . That is to say: if as both n and m independently go to infinity, then there is some with .
Every Euclidean space is complete, as is every closed subset of a complete space. The rational numbers, using the absolute value metric d(x,y)=|x-y|, are not complete.
Every metric space has a unique completion, which is a complete space that contains the given space as a dense subset. For example, the real numbers are the completion of the rationals.
If X is a complete subset of the metric space M, then X is closed in M. Indeed, a space is complete if and only if it is closed in any containing metric space.
Every complete metric space is a Baire space.
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 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 intervals in the space of real numbers and occasionally regions in a Euclidean space R^{n} 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 that converges to a point in M. This is known as sequential compactness and, in metric spaces (but not in general topological spaces), is equivalent to the topological notions of countable compactness and compactness 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. Every closed subset of a compact space is itself compact.
A metric space is compact if and only if it is complete and totally bounded. This is known as the Heine–Borel theorem. Note that compactness depends only on the topology, while boundedness depends on the metric.
Lebesgue's number lemma 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 < δ is contained in some member of the cover.
Every compact metric space is countable,^{[1]} and is a continuous image of the Cantor set. (The latter result is due to Pavel Alexandrov and Urysohn.)
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 M is connected if the only subsets that are both open and closed are the empty set and M itself.
A metric space M is path connected if two any two points x, y in M there exists a continuous map f:[0,1]→M with f(0)=x and f(1)=y. Every path connected space is connected, but the converse is not true in general.
There are also local versions of these definitions: locally connected spaces and locally path connected spaces.
Simply connected spaces 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 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 property.
Types of maps between metric spaces
Suppose (M_{1},d_{1}) and (M_{2},d_{2}) are two metric spaces.
Continuous maps
The map f:M_{1}→M_{2} is continuous if it has one (and therefore all) of the following equivalent properties:
- general topological continuity
- for every open set U in M_{2}, the preimage f^{ -1}(U) is open in M_{1}
- This is the general definition of continuity in topology.
- sequential continuity
- if (x_{n}) is a sequence in M_{1} that converges to x in M_{1}, then the sequence (f(x_{n})) converges to f(x) in M_{2}.
- This is sequential continuity, due to Eduard Heine.
- ε-δ definition
- for every x in M_{1} and every ε>0 there exists δ>0 such that
- This uses the (ε, δ)-definition of limit, and is due to Augustin Louis Cauchy.
Moreover, f is continuous if and only if it is continuous on every compact subset of M_{1}.
The image 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 : M_{1} → M_{2} is uniformly continuous if for every ε > 0 there exists δ > 0 such that
Every uniformly continuous map ƒ : M_{1} → M_{2} is continuous. The converse is true if M_{1} is compact (Heine–Cantor theorem).
Uniformly continuous maps turn Cauchy sequences in M_{1} into Cauchy sequences into M_{2}.
Lipschitz-continuous maps and contractions
Given a number K > 0, the map ƒ : M_{1} → M_{2} is K-Lipschitz continuous 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. Suppose M_{2} = M_{1} and M_{1} is complete. If ƒ is a contraction, then ƒ admits a unique fixed point (Banach fixed point theorem). If M_{1} is compact, the condition can be weakened a bit: ƒ admits a unique fixed point if
for all .
Isometries
The map f:M_{1}→M_{2} is an isometry 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:M_{1}→M_{2} is a quasi-isometry if there exist constants A≥1 and B≥0 such that
and a constant C≥0 such that every point in M_{2} has a distance at most C from some point in the image f(M_{1}).
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 in relation to the word metric.
Notions of metric space equivalence
Given two metric spaces (M_{1}, d_{1}) and (M_{2}, d_{2}):
- 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 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^{[2]} Hausdorff spaces^{[3]} 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.
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 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 is continuous.
Given two subsets S and T of M, we define their Hausdorff distance to be
- d_{H}(S,T) = max { sup {d(s,T) : s ∈ S}, sup {d(t,S) : t ∈ T} }
In general, the Hausdorff distance d_{H}(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 d_{H} 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.)
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 R^{n}, then is a metric space, where the product metric is defined by
- ,
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
- .
An uncountable product of metric spaces need not be metrizable. For example, is not first-countable 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, 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 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 and with , , . In general this will only define a pseudometric, i.e. 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 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 if and only if it is a quotient of a metric space.^{[4]}
Generalizations of metric spaces
Every metric space is a uniform space in a natural manner, and every uniform space is naturally a topological space. 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, a quasimetric space, or a semimetric space.
If the distance function takes values in the extended real number line R∪{+∞}, but otherwise satisfies all four conditions, then it is called an extended metric and the corresponding space is called an -metric space.
See also
- Glossary of Riemannian and metric geometry
- Topology
- Triangle inequality
- Lipschitz continuity
- isometry, contraction mapping and metric map
- Category of metric spaces
- Measure (mathematics)
- Norm (mathematics)
- Basic introduction to the mathematics of curved spacetime
- Classical Wiener space
- Premetric space
- Space (mathematics)
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, from PlanetMath.org. Accessed 10 July 2010.
- ↑ Goreham, Anthony. Sequential convergence in Topological Spaces. Honours' Dissertation, Queen's College, Oxford (April, 2001), p. 14
References
- 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.
External links
- Far and near — several examples of distance functions at cut-the-knot