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 segment connecting them. Other metric spaces occur for example in elliptic geometry and hyperbolic geometry, where distance on a sphere measured by angle is a metric, and the hyperboloid model of hyperbolic geometry is used by special relativity as a metric space of velocities.

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

History

edit

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

Definition

edit

A metric space is an ordered pair   where   is a set and   is a metric on  , i.e., a function

 

such that for any  , the following holds:

  1.       (non-negative),
  2.   iff       (identity of indiscernibles),
  3.       (symmetry) and
  4.       (triangle inequality) .

The first condition follows from the other three, since:

 

The function   is also called distance function or simply distance. Often,   is omitted and one just writes   for a metric space if it is clear from the context what metric is used.

Examples of metric spaces

edit
  • 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   given by the absolute difference, and more generally Euclidean  -space with the Euclidean distance, are complete metric spaces. The rational numbers with the same distance also form a metric space, but are not complete.
  • The positive real numbers with distance function   is a complete metric space.
  • Any normed vector space is a metric space by defining  , see also metrics on vector spaces. (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 differences 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   to  .
  • The British Rail metric (also called the Post Office metric or the SNCF metric) on a normed vector space is given by   for distinct points   and  , and  . More generally   can be replaced with a function   taking an arbitrary set   to non-negative reals and taking the value   at most once: then the metric is defined on   by   for distinct points   and  , and  . The name alludes to the tendency of railway journeys (or letters) to proceed via London (or Paris) irrespective of their final destination.
  • If   is a metric space and   is a subset of  , then   becomes a metric space by restricting the domain of   to  .
  • The discrete metric, where   if   and   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.
  • A finite metric space is a metric space having a finite number of points. Not every finite metric space can be isometrically embedded in a Euclidean space.[1][2]
  • The hyperbolic plane is a metric space. More generally:
  • If   is any connected Riemannian manifold, then we can turn   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   is some set and   is a metric space, then, the set of all bounded functions   (i.e. those functions whose image is a bounded subset of  ) can be turned into a metric space by defining   for any two bounded functions   and   (where   is supremum.[3] This metric is called the uniform metric or supremum metric, and If   is complete, then this function space is complete as well. If X is also a topological space, then the set of all bounded continuous functions from   to   (endowed with the uniform metric), will also be a complete metric if M is.
  • If   is an undirected connected graph, then the set   of vertices of   can be turned into a metric space by defining   to be the length of the shortest path connecting the vertices   and  . 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   and  , defined as the minimal number of character deletions, insertions, or substitutions required to transform   into  . This can be thought of as a special case of the shortest path metric in a graph and is one example of an edit distance.
  • Given a metric space   and an increasing concave function   such that   if and only if  , then   is also a metric on  .
  • Given an injective function   from any set   to a metric space  ,   defines a metric on  .
  • 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   by   matrices over some field is a metric space with respect to the rank distance  .
  • The Helly metric is used in game theory.

Open and closed sets, topology and convergence

edit

Every metric space is a topological space in a natural manner, and therefore all definitions and theorems about general topological spaces also apply to all metric spaces.

About any point   in a metric space   we define the open ball of radius   about   as the set

 

These open balls form the base for a topology on M, making it a topological space.

Explicitly, a subset   of   is called open if for every   in   there exists an   such that   is contained in  . The complement of an open set is called closed. A neighborhood of the point   is any subset of   that contains an open ball about   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 ( ) in a metric space   is said to converge to the limit   iff for every  , there exists a natural number N such that   for all  . Equivalently, one can use the general definition of convergence available in all topological spaces.

A subset   of the metric space   is closed iff every sequence in   that converges to a limit in   has its limit in  .

Types of metric spaces

edit

Complete spaces

edit

A metric space   is said to be complete if every Cauchy sequence converges in  . That is to say: if   as both   and   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  , are not complete.

Every metric space has a unique (up to isometry) 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   is a complete subset of the metric space  , then   is closed in  . Indeed, a space is complete iff it is closed in any containing metric space.

Every complete metric space is a Baire space.

Bounded and totally bounded spaces

edit
 
Diameter of a set.

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 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.

Compact spaces

edit

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 iff 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 second countable,[4] and is a continuous image of the Cantor set. (The latter result is due to Pavel Alexandrov and Urysohn.)

Locally compact and proper spaces

edit

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

edit

A metric space   is connected if the only subsets that are both open and closed are the empty set and   itself.

A metric space   is path connected if for any two points   there exists a continuous map   with   and  . 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".