Metric space
In
mathematics, a
metric space is a
set (or "space") where a distance between points is defined.
Maurice Fréchet introduced metric spaces in his work Sur quelques points du calcul fonctionnel in 1906.
Formally, a metric space M is a set of points with an associated distance function (also called a metric)
d : M × M -> R (where R is the set of real numbers). For all x, y, z in M, this function is required to satisfy the following conditions:
- d(x, y) ≥ 0
- d(x, x) = 0
- if d(x, y) = 0 then x = y (identity of indiscernibles)
- d(x, y) = d(y, x) (symmetry)
- d(x, z) ≤ d(x, y) + d(y, z) (triangle inequality).
These axioms express intuitive notions about the concept of "distance": distances between different spots are positive and the distance between
x and
y is the same as the distance between
y and
x. The triangle inequality means that if you go from
x to
z directly, that is no longer than going first from
x to
y, and then from
y to
z. In
Euclidean geometry, this is easy to see. Metric spaces allow this concept to be extended to a more abstract setting.
In metric spaces, one can talk about limits of sequences; a metric space in which every Cauchy sequence has a limit is said to be complete.
- The trivial distance metric: d(x,y) = 0 if x = y else 1.
- 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.
- More generally still, any normed vector space is a metric space by defining d(x, y) = ||y - x||. If such a space is complete, we call it a Banach 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.
- 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.
Further definitions and properties
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.
A metric space M is called bounded if there exists some number r > 0 such that d(x,y) ≤ r for all x and y in M (not to be confused with "finite", which refers to the number of elements, not to how far the set extends; finiteness implies boundedness, but not conversely). The space M is called totally bounded if for every r > 0 there exist finitely many open balls of radius r whose union equals M. It is not difficult to see that every totally bounded space is bounded. It can be shown that a metric space is compact if and only if it is complete and totally bounded.
By restricting the metric, any subset of a metric space is a metric space itself. We call such a subset complete, bounded, totally bounded or compact if it, considered as a metric space, has the corresponding property.
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.
An isometry between two metric spaces (M_{1}, d_{1}) and (M_{2}, d_{2}) is a function f : M_{1} → M_{2} with the property d_{2}(f(x), f(y)) = d_{1}(x, y) for all x, y in M_{1}. Isometries are necessarily injective. We call two spaces isometrically isomorphic if there exists a bijective isometry between them. In this case, the two spaces are essentially identical.
Every metric space is isometrically isomorphic to a closed subset of some normed vector space. Every complete metric space is isometrically isomorphic to a closed subset of some Banach space.
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 |-> d(
x,
S) is continuous.
The property 1 (d(x, y) ≥ 0) follows from properties 2, 4 and 5 and does not have to be required separately.
Some authors use the extended real number line and allow the distance function d to attain the value ∞. Every such metric can be rescaled to a finite metric (using d'(x, y) = d(x, y) / (1 + d(x, y)) or d''(x, y) = min(1, d(x, y))) and the two concepts of metric space are therefore equivalent as far as notions of topology (such as continuity or convergence) are concerned.
A metric is called an ultrametric if it satisfies the following stronger version of the triangle inequality:
- For all x, y, z in M, d(x, z) ≤ max(d(x, y), d(y, z))
If one drops property 3, one obtains pseudometric spaces. Dropping property 4 instead, one obtains quasimetric spaces. However, losing symmetry in this case, one usually changes property 3 such that both
d(
x,
y)=0 and
d(
y,
x)=0 are needed for
x and
y to be identified. All combinations of the above are possible and are referred to by their according names (such as
quasi-pseudo-ultrametric).