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Mathematical set defining distance
In mathematics, a metric space is a set together with a metric on the set. The metric is a function that defines a concept of distance between any two members of the set, which are usually called points. The metric satisfies a few simple properties. Informally:
the distance from to is zero if and only if and are the same point,
the distance between two distinct points is positive,
the distance from to is the same as the distance from to , and
the distance from to (directly) is less than or equal to the distance from to via any third point .
Given the above three axioms, we also have that for any . This is deduced as follows:
by triangle inequality
by identity of indiscernibles
we have non-negativity
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.
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. If the distance between two points is zero, the two points are indistinguishable from one-another. Many of the examples below can be seen as concrete versions of this general idea.
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 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 sets. This, in particular, shows that for any 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.
If is any connectedRiemannian 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). 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.
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 .
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.
A sequence () in a metric space is said to converge to the limit if and only if 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 if and only if every sequence in that converges to a limit in has its limit in .
Types of metric spaces
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 if and only if it is closed in any containing metric space.
A metric space is called bounded if there exists some number , such that for all . The smallest possible such is called the diameter of . The space is called precompact or totally bounded if for every there exist finitely many open balls of radius whose union covers . 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 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. Also note that an unbounded subset of may have a finite volume.
Examples of compact metric spaces include the closed interval 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 , there exists a "Lebesgue number" such that every subset of of diameter is contained in some member of the cover.
If is a nonempty metric space and then is called a pointed metric space, and is called a distinguished point. Note that a pointed metric space is just a nonempty metric space with attention drawn to its distinguished point, and that any nonempty metric space can be viewed as a pointed metric space. The distinguished point is sometimes denoted due to its similar behavior to zero in certain contexts.
Types of maps between metric spaces
Suppose and are two metric spaces.
The map is continuous
if it has one (and therefore all) of the following equivalent properties:
Every uniformly continuous map is continuous. The converse is true if is compact (Heine-Cantor theorem).
Uniformly continuous maps turn Cauchy sequences in into Cauchy sequences in . For continuous maps this is generally wrong; for example, a continuous map
from the open interval onto the real line turns some Cauchy sequences into unbounded sequences.
Every Lipschitz-continuous map is uniformly continuous, but the converse is not true in general.
If , then is called a contraction. Suppose and is complete. If is a contraction, then admits a unique fixed point (Banach fixed-point theorem). If is compact, the condition can be weakened a bit: admits a unique fixed point 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).
and a constant such that every point in has a distance at most from some point in the image .
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 and :
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 bijectiveisometry 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.
Metric spaces are paracompactHausdorff 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.
Metric spaces are first countable since one can use balls with rational radius as a neighborhood base.
The metric topology on a metric space is the coarsest topology on relative to which the metric is a continuous map from the product of with itself to the non-negative real numbers.
Distance between points and sets; Hausdorff distance and Gromov metric
In general, the Hausdorff distance 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 turns the set of all non-empty compact subsets of into a metric space. One can show that is complete if 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 class of all (isometry classes of) compact metric spaces becomes a metric space in its own right.
and the induced topology agrees with the product topology. By the equivalence of norms in finite dimensions, an equivalent metric is obtained if is the taxicab norm, a p-norm, the maximum norm, or any other norm which is non-decreasing as the coordinates of a positive -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
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 is continuous with respect to the product topology of .
Quotient metric spaces
If M is a metric space with metric , and is an equivalence relation on , then we can endow the quotient set with a pseudometric. Given two equivalence classes and , 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 . However, for some equivalence relations (e.g., those given by gluing together polyhedra along faces), is a metric.
The quotient metric is characterized by the following universal property. If is a metric map between metric spaces (that is, for all , ) satisfying 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.
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 the distance function takes values in the extended real number line, but otherwise satisfies all four conditions, then it is called an extended metric and the corresponding space is called an -metric space. If the distance function takes values in some (suitable) ordered set (and the triangle inequality is adjusted accordingly), then we arrive at the notion of generalized ultrametric.
Approach spaces are a generalization of metric spaces, based on point-to-set distances, instead of point-to-point distances.