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{{Short description|Mathematical space with a notion of distance}}{{use dmy dates|date=December 2020|cs1-dates=y}}File:Manhattan distance.svg|thumb|200px|The plane (a set of points) can be equipped with different metrics. In the taxicab metric the red, yellow and blue paths have the same length (12), and are all shortest paths. In the Euclidean metricEuclidean metricIn mathematics, a metric space is a set together with a notion of distance between its elements, usually called points. The distance is measured by a function called a metric or distance function.{{sfn|Čech|1969|p=42}} Metric spaces are the most general setting for studying many of the concepts of mathematical analysis and geometry.The most familiar example of a metric space is 3-dimensional Euclidean space with its usual notion of distance. Other well-known examples are a sphere equipped with the angular distance and the hyperbolic plane. A metric may correspond to a metaphorical, rather than physical, notion of distance: for example, the set of 100-character Unicode strings can be equipped with the Hamming distance, which measures the number of characters that need to be changed to get from one string to another.Since they are very general, metric spaces are a tool used in many different branches of mathematics. Many types of mathematical objects have a natural notion of distance and therefore admit the structure of a metric space, including Riemannian manifolds, normed vector spaces, and graphs. In abstract algebra, the p-adic numbers arise as elements of the completion of a metric structure on the rational numbers. Metric spaces are also studied in their own right in metric geometry{{sfn|Burago|Burago|Ivanov|2001}} and analysis on metric spaces.{{sfn|Heinonen|2001}}Many of the basic notions of mathematical analysis, including balls, completeness, as well as uniform, Lipschitz, and Hölder continuity, can be defined in the setting of metric spaces. Other notions, such as continuity, compactness, and open and closed sets, can be defined for metric spaces, but also in the even more general setting of topological spaces.

Definition and illustration

Motivation

(File:Great-circle distance vs straight line distance.svg|thumb|A diagram illustrating the great-circle distance (in cyan) and the straight-line distance (in red) between two points {{mvar|P}} and {{mvar|Q}} on a sphere.)To see the utility of different notions of distance, consider the surface of the Earth as a set of points. We can measure the distance between two such points by the length of the shortest path along the surface, “as the crow flies”; this is particularly useful for shipping and aviation. We can also measure the straight-line distance between two points through the Earth’s interior; this notion is, for example, natural in seismology, since it roughly corresponds to the length of time it takes for seismic waves to travel between those two points.The notion of distance encoded by the metric space axioms has relatively few requirements. This generality gives metric spaces a lot of flexibility. At the same time, the notion is strong enough to encode many intuitive facts about what distance means. This means that general results about metric spaces can be applied in many different contexts.Like many fundamental mathematical concepts, the metric on a metric space can be interpreted in many different ways. A particular metric may not be best thought of as measuring physical distance, but, instead, as the cost of changing from one state to another (as with Wasserstein metrics on spaces of measures) or the degree of difference between two objects (for example, the Hamming distance between two strings of characters, or the Gromov–Hausdorff distance between metric spaces themselves).

Definition

Formally, a metric space is an ordered pair {{math|(M, d)}} where {{mvar|M}} is a set and {{mvar|d}} is a metric on {{mvar|M}}, i.e., a functiond,colon M times M to mathbb{R}satisfying the following axioms for all points x,y,z in M:{{sfn|Burago|Burago|Ivanov|2001|p=1}}{{sfn|Gromov|2007|p=xv}}

1. The distance from a point to itself is zero: d(x, x) = 0
2. (Positivity) The distance between two distinct points is always positive: text{If }x neq ytext{, then }d(x, y)>0
3. (Symmetry) The distance from {{mvar|x}} to {{mvar|y}} is always the same as the distance from {{mvar|y}} to {{mvar|x}}: d(x, y) = d(y, x)
4. The triangle inequality holds: d(x, z) leq d(x, y) + d(y, z)This is a natural property of both physical and metaphorical notions of distance: you can arrive at {{mvar|z}} from {{mvar|x}} by taking a detour through {{mvar|y}}, but this will not make your journey any shorter than the direct path.

If the metric {{mvar|d}} is unambiguous, one often refers by abuse of notation to “the metric space {{mvar|M}}”.By taking all axioms except the second, one can show that distance is always non-negative:0 = d(x, x) leq d(x, y) + d(y, x) = 2 d(x, y)Therefore the second axiom can be weakened to text{If }x neq ytext{, then }d(x, y) neq 0 and combined with the first to make d(x, y) = 0 iff x=y.BOOK, Gleason, Andrew, Fundamentals of Abstract Analysis, Taylor & Francis, 1991, 1st, 223, 10.1201/9781315275444, 9781315275444, 62222843,

Simple examples

The real numbers

The real numbers with the distance function d(x,y) = | y - x | given by the absolute difference form a metric space. Many properties of metric spaces and functions between them are generalizations of concepts in real analysis and coincide with those concepts when applied to the real line.

Metrics on Euclidean spaces

(File:Minkowski_distance_examples.svg|thumb|Comparison of Chebyshev, Euclidean and taxicab distances for the hypotenuse of a 3-4-5 triangle on a chessboard)The Euclidean plane R^2 can be equipped with many different metrics. The Euclidean distance familiar from school mathematics can be defined byd_2((x_1,y_1),(x_2,y_2))=sqrt{(x_2-x_1)^2+(y_2-y_1)^2}.The taxicab or Manhattan distance is defined byd_1((x_1,y_1),(x_2,y_2))=|x_2-x_1|+|y_2-y_1|and can be thought of as the distance you need to travel along horizontal and vertical lines to get from one point to the other, as illustrated at the top of the article.The maximum, L^infty, or Chebyshev distance is defined byd_infty((x_1,y_1),(x_2,y_2))=max{|x_2-x_1|,|y_2-y_1|}.This distance does not have an easy explanation in terms of paths in the plane, but it still satisfies the metric space axioms. It can be thought of similarly to the number of moves a king would have to make on a chess board to travel from one point to another on the given space. In fact, these three distances, while they have distinct properties, are similar in some ways. Informally, points that are close in one are close in the others, too. This observation can be quantified with the formulad_infty(p,q) leq d_2(p,q) leq d_1(p,q) leq 2d_infty(p,q),which holds for every pair of points p, q in R^2.A radically different distance can be defined by settingd(p,q)=begin{cases}0, & text{if }p=q, 1, & text{otherwise.}end{cases}Using Iverson brackets,d(p,q) = [pne q]In this discrete metric, all distinct points are 1 unit apart: none of them are close to each other, and none of them are very far away from each other either. Intuitively, the discrete metric no longer remembers that the set is a plane, but treats it just as an undifferentiated set of points.All of these metrics make sense on R^n as well as R^2.

Subspaces

Given a metric space {{math|(M, d)}} and a subset A subseteq M, we can consider {{mvar|A}} to be a metric space by measuring distances the same way we would in {{mvar|M}}. Formally, the induced metric on {{mvar|A}} is a function d_A:A times A to R defined byd_A(x,y)=d(x,y).For example, if we take the two-dimensional sphere {{math|S2}} as a subset of R^3, the Euclidean metric on R^3 induces the straight-line metric on {{math|S2}} described above. Two more useful examples are the open interval {{open-open|0, 1}} and the closed interval {{closed-closed|0, 1}} thought of as subspaces of the real line.

History

{{Expand section|Reasons for generalizing the Euclidean metric, first non-Euclidean metrics studied, consequences for mathematics|date=August 2011}}In 1906 René Maurice Fréchet introduced metric spaces in his work Sur quelques points du calcul fonctionnelJOURNAL, Fréchet, M., Sur quelques points du calcul fonctionnel, Rendiconti del Circolo Matematico di Palermo, December 1906, 22, 1, 1–72, 10.1007/BF03018603, 123251660,zenodo.org/record/1428464, in the context of functional analysis: his main interest was in studying the real-valued functions from a metric space, generalizing the theory of functions of several or even infinitely many variables, as pioneered by mathematicians such as Cesare Arzelà. The idea was further developed and placed in its proper context by Felix Hausdorff in his magnum opus Principles of Set Theory, which also introduced the notion of a (Hausdorff) topological space.JOURNAL, Blumberg, Henry, Hausdorff’s Grundzüge der Mengenlehre, Bulletin of the American Mathematical Society, 1927, 6, 778–781, 10.1090/S0002-9904-1920-03378-1, free, General metric spaces have become a foundational part of the mathematical curriculum.{{sfn|Rudin|1976|p=30}} Prominent examples of metric spaces in mathematical research include Riemannian manifolds and normed vector spaces, which are the domain of differential geometry and functional analysis, respectively.E.g. {{harvnb|Burago|Burago|Ivanov|2001|p=xiii}}: ... for most of the last century it was a common belief that “geometry of manifolds” basically boiled down to “analysis on manifolds”. Geometric methods heavily relied on differential machinery, as can be guessed from the name “Differential geometry”. Fractal geometry is a source of some exotic metric spaces. Others have arisen as limits through the study of discrete or smooth objects, including scale-invariant limits in statistical physics, Alexandrov spaces arising as Gromov–Hausdorff limits of sequences of Riemannian manifolds, and boundaries and asymptotic cones in geometric group theory. Finally, many new applications of finite and discrete metric spaces have arisen in computer science.

Basic notions

A distance function is enough to define notions of closeness and convergence that were first developed in real analysis. Properties that depend on the structure of a metric space are referred to as metric properties. Every metric space is also a topological space, and some metric properties can also be rephrased without reference to distance in the language of topology; that is, they are really topological properties.

The topology of a metric space

For any point {{mvar|x}} in a metric space {{mvar|M}} and any real number {{math|r > 0}}, the open ball of radius {{mvar|r}} around {{mvar|x}} is defined to be the set of points that are strictly less than distance {{mvar|r}} from {{mvar|x}}:B_r(x)={y in M : d(x,y) < r}.This is a natural way to define a set of points that are relatively close to {{mvar|x}}. Therefore, a set N subseteq M is a neighborhood of {{mvar|x}} (informally, it contains all points “close enough” to {{mvar|x}}) if it contains an open ball of radius {{mvar|r}} around {{mvar|x}} for some {{math|r > 0}}.An open set is a set which is a neighborhood of all its points. It follows that the open balls form a base for a topology on {{mvar|M}}. In other words, the open sets of {{mvar|M}} are exactly the unions of open balls. As in any topology, closed sets are the complements of open sets. Sets may be both open and closed as well as neither open nor closed.This topology does not carry all the information about the metric space. For example, the distances {{math|d1}}, {{math|d2}}, and {{math|d∞}} defined above all induce the same topology on R^2, although they behave differently in many respects. Similarly, R with the Euclidean metric and its subspace the interval {{open-open|0, 1}} with the induced metric are homeomorphic but have very different metric properties.Conversely, not every topological space can be given a metric. Topological spaces which are compatible with a metric are called metrizable and are particularly well-behaved in many ways: in particular, they are paracompactRudin, Mary Ellen. A new proof that metric spaces are paracompact {{webarchive|url=https://web.archive.org/web/20160412015215www.jstor.org/stable/2035708 |date=2016-04-12 }}. Proceedings of the American Mathematical Society, Vol. 20, No. 2. (Feb., 1969), p. 603. Hausdorff spaces (hence normal) and first-countable.{{efn|Balls with rational radius around a point {{mvar|x}} form a neighborhood basis for that point.}} The Nagata–Smirnov metrization theorem gives a characterization of metrizability in terms of other topological properties, without reference to metrics.

Convergence

Convergence of sequences in Euclidean space is defined as follows: Convergence of sequences in a topological space is defined as follows: In metric spaces, both of these definitions make sense and they are equivalent. This is a general pattern for topological properties of metric spaces: while they can be defined in a purely topological way, there is often a way that uses the metric which is easier to state or more familiar from real analysis.

Completeness

Informally, a metric space is complete if it has no “missing points”: every sequence that looks like it should converge to something actually converges.To make this precise: a sequence {{math|(xn)}} in a metric space {{mvar|M}} is Cauchy if for every {{math|ε > 0}} there is an integer {{mvar|N}} such that for all {{math|m, n > N}}, {{math|d(xm, xn) < ε}}. By the triangle inequality, any convergent sequence is Cauchy: if {{mvar|xm}} and {{mvar|xn}} are both less than {{math|ε}} away from the limit, then they are less than {{math|2ε}} away from each other. If the converse is true—every Cauchy sequence in {{mvar|M}} converges—then {{mvar|M}} is complete.Euclidean spaces are complete, as is R^2 with the other metrics described above. Two examples of spaces which are not complete are {{open-open|0, 1}} and the rationals, each with the metric induced from R. One can think of {{open-open|0, 1}} as “missing” its endpoints 0 and 1. The rationals are missing all the irrationals, since any irrational has a sequence of rationals converging to it in R (for example, its successive decimal approximations). These examples show that completeness is not a topological property, since R is complete but the homeomorphic space {{open-open|0, 1}} is not.This notion of “missing points” can be made precise. In fact, every metric space has a unique completion, which is a complete space that contains the given space as a dense subset. For example, {{closed-closed|0, 1}} is the completion of {{open-open|0, 1}}, and the real numbers are the completion of the rationals.Since complete spaces are generally easier to work with, completions are important throughout mathematics. For example, in abstract algebra, the p-adic numbers are defined as the completion of the rationals under a different metric. Completion is particularly common as a tool in functional analysis. Often one has a set of nice functions and a way of measuring distances between them. Taking the completion of this metric space gives a new set of functions which may be less nice, but nevertheless useful because they behave similarly to the original nice functions in important ways. For example, weak solutions to differential equations typically live in a completion (a Sobolev space) rather than the original space of nice functions for which the differential equation actually makes sense.

Bounded and totally bounded spaces

(File:Diameter of a Set.svg|thumb|Diameter of a set.){{See also|Bounded set}}A metric space {{mvar|M}} is bounded if there is an {{mvar|r}} such that no pair of points in {{mvar|M}} is more than distance {{mvar|r}} apart.{{efn|In the context of intervals in the real line, or more generally regions in Euclidean space, bounded sets are sometimes referred to as “finite intervals” or “finite regions”. However, they do not typically have a finite number of elements, and while they all have finite volume, so do many unbounded sets. Therefore this terminology is imprecise.}} The least such {{mvar|r}} is called the {{visible anchor|diameter|Diameter of a metric space}} of {{mvar|M}}.The space {{mvar|M}} is called precompact or totally bounded if for every {{math|r > 0}} there is a finite cover of {{mvar|M}} by open balls of radius {{mvar|r}}. Every totally bounded space is bounded. To see this, start with a finite cover by {{mvar|r}}-balls for some arbitrary {{mvar|r}}. Since the subset of {{mvar|M}} consisting of the centers of these balls is finite, it has finite diameter, say {{mvar|D}}. By the triangle inequality, the diameter of the whole space is at most {{math|D + 2r}}. The converse does not hold: an example of a metric space that is bounded but not totally bounded is R^2 (or any other infinite set) with the discrete metric.

Compactness

Compactness is a topological property which generalizes the properties of a closed and bounded subset of Euclidean space. There are several equivalent definitions of compactness in metric spaces:

1. A metric space {{mvar|M}} is compact if every open cover has a finite subcover (the usual topological definition).
2. A metric space {{mvar|M}} is compact if every sequence has a convergent subsequence. (For general topological spaces this is called sequential compactness and is not equivalent to compactness.)
3. A metric space {{mvar|M}} is compact if it is complete and totally bounded. (This definition is written in terms of metric properties and does not make sense for a general topological space, but it is nevertheless topologically invariant since it is equivalent to compactness.)

One example of a compact space is the closed interval {{closed-closed|0, 1}}.Compactness is important for similar reasons to completeness: it makes it easy to find limits. Another important tool is Lebesgue’s number lemma, which shows that for any open cover of a compact space, every point is relatively deep inside one of the sets of the cover.

Functions between metric spaces

File:Functions between metric spaces.svg|thumb|upright=1.25|Euler diagramEuler diagramUnlike in the case of topological spaces or algebraic structures such as groups or rings, there is no single “right” type of structure-preserving function between metric spaces. Instead, one works with different types of functions depending on one’s goals. Throughout this section, suppose that (M_1,d_1) and (M_2,d_2) are two metric spaces. The words “function” and “map” are used interchangeably.

Isometries

One interpretation of a “structure-preserving” map is one that fully preserves the distance function: It follows from the metric space axioms that a distance-preserving function is injective. A bijective distance-preserving function is called an isometry.{{harvnb|Burago|Burago|Ivanov|2001|p=2}}.Some authors refer to any distance-preserving function as an isometry, e.g. {{harvnb|Munkres|2000|p=181}}. One perhaps non-obvious example of an isometry between spaces described in this article is the map f:(R^2,d_1) to (R^2,d_infty) defined byf(x,y)=(x+y,x-y).If there is an isometry between the spaces {{math|M1}} and {{math|M2}}, they are said to be isometric. Metric spaces that are isometric are essentially identical.

Continuous maps

On the other end of the spectrum, one can forget entirely about the metric structure and study continuous maps, which only preserve topological structure. There are several equivalent definitions of continuity for metric spaces. The most important are:

• Topological definition. A function f,colon M_1to M_2 is continuous if for every open set {{mvar|U}} in {{math|M2}}, the preimage f^{-1}(U) is open.
• Sequential continuity. A function f,colon M_1to M_2 is continuous if whenever a sequence {{math|(xn)}} converges to a point {{mvar|x}} in {{math|M1}}, the sequence f(x_1),f(x_2),ldots converges to the point {{math|f(x)}} in {{math|M2}}.

• ε–δ definition. A function f,colon M_1to M_2 is continuous if for every point {{mvar|x}} in {{math|M1}} and every {{math|ε > 0}} there exists {{math|δ > 0}} such that for all {{mvar|y}} in {{math|M1}} we have d_1(x,y) < delta implies d_2(f(x),f(y)) < varepsilon.

A homeomorphism is a continuous bijection whose inverse is also continuous; if there is a homeomorphism between {{math|M1}} and {{math|M2}}, they are said to be homeomorphic. Homeomorphic spaces are the same from the point of view of topology, but may have very different metric properties. For example, R is unbounded and complete, while {{open-open|0, 1}} is bounded but not complete.

Uniformly continuous maps

A function f,colon M_1to M_2 is uniformly continuous if for every real number {{math|ε > 0}} there exists {{math|δ > 0}} such that for all points {{mvar|x}} and {{mvar|y}} in {{math|M1}} such that d(x,y) 0}}, the map f,colon M_1to M_2 is {{mvar|K}}-Lipschitz ifd_2(f(x),f(y))leq K d_1(x,y)quadtext{for all}quad x,yin M_1.Lipschitz maps are particularly important in metric geometry, since they provide more flexibility than distance-preserving maps, but still make essential use of the metric.{{sfn|Gromov|2007|p=xvii}} For example, a curve in a metric space is rectifiable (has finite length) if and only if it has a Lipschitz reparametrization.A 1-Lipschitz map is sometimes called a nonexpanding or metric map. Metric maps are commonly taken to be the morphisms of the category of metric spaces.A {{mvar|K}}-Lipschitz map for {{math|K < 1}} is called a contraction. The Banach fixed-point theorem states that if {{mvar|M}} is a complete metric space, then every contraction f:M to M admits a unique fixed point. If the metric space {{mvar|M}} is compact, the result holds for a slightly weaker condition on {{mvar|f}}: a map f:M to M admits a unique fixed point if

Quasi-isometries

A quasi-isometry is a map that preserves the “large-scale structure” of a metric space. Quasi-isometries need not be continuous. For example, R^2 and its subspace Z^2 are quasi-isometric, even though one is connected and the other is discrete. The equivalence relation of quasi-isometry is important in geometric group theory: the Å varc–Milnor lemma states that all spaces on which a group acts geometrically are quasi-isometric.{{sfn|Margalit|Thomas|2017}}Formally, the map f,colon M_1to M_2 is a quasi-isometric embedding if there exist constants {{math|A ≥ 1}} and {{math|B ≥ 0}} such thatfrac{1}{A} d_2(f(x),f(y))-Bleq d_1(x,y)leq A d_2(f(x),f(y))+B quadtext{ for all }quad x,yin M_1.It is a quasi-isometry if in addition it is quasi-surjective, i.e. there is a constant {{math|C ≥ 0}} such that every point in M_2 is at distance at most {{mvar|C}} from some point in the image f(M_1).

Notions of metric space equivalence

{{See also|Equivalence of metrics}}Given two metric spaces (M_1, d_1) and (M_2, d_2):

• They are called homeomorphic (topologically isomorphic) if there is a homeomorphism between them (i.e., a continuous bijection with a continuous inverse). If M_1=M_2 and the identity map is a homeomorphism, then d_1 and d_2 are said to be topologically equivalent.
• They are called uniformic (uniformly isomorphic) if there is a uniform isomorphism between them (i.e., a uniformly continuous bijection with a uniformly continuous inverse).
• They are called bilipschitz homeomorphic if there is a bilipschitz bijection between them (i.e., a Lipschitz bijection with a Lipschitz inverse).
• They are called isometric if there is a (bijective) isometry between them. In this case, the two metric spaces are essentially identical.
• They are called quasi-isometric if there is a quasi-isometry between them.

Metric spaces with additional structure

Normed vector spaces

{{anchor|Norm induced metric|Relation of norms and metrics}}A normed vector space is a vector space equipped with a norm, which is a function that measures the length of vectors. The norm of a vector {{mvar|v}} is typically denoted by lVert v rVert. Any normed vector space can be equipped with a metric in which the distance between two vectors {{mvar|x}} and {{mvar|y}} is given byd(x,y)=lVert x-y rVert.The metric {{mvar|d}} is said to be induced by the norm lVert{cdot}rVert. Conversely,{{sfn|Narici|Beckenstein|2011|pp=47–66}} if a metric {{mvar|d}} on a vector space {{mvar|X}} is

• translation invariant: d(x,y) = d(x+a,y+a) for every {{mvar|x}}, {{mvar|y}}, and {{mvar|a}} in {{mvar|X}}; and
• {{visible anchor|absolutely homogeneous|Homogeneous metric}}: d(alpha x, alpha y) = |alpha| d(x,y) for every {{mvar|x}} and {{mvar|y}} in {{mvar|X}} and real number {{math|α}};

then it is the metric induced by the normlVert x rVert = d(x,0).A similar relationship holds between seminorms and pseudometrics.Among examples of metrics induced by a norm are the metrics {{math|d1}}, {{math|d2}}, and {{math|d∞}} on R^2, which are induced by the Manhattan norm, the Euclidean norm, and the maximum norm, respectively. More generally, the Kuratowski embedding allows one to see any metric space as a subspace of a normed vector space.Infinite-dimensional normed vector spaces, particularly spaces of functions, are studied in functional analysis. Completeness is particularly important in this context: a complete normed vector space is known as a Banach space. An unusual property of normed vector spaces is that linear transformations between them are continuous if and only if they are Lipschitz. Such transformations are known as bounded operators.

Length spaces

File:approximate arc length.svg|thumb|One possible approximation for the arc length of a curve. The approximation is never longer than the arc length, justifying the definition of arc length as a supremumsupremumA curve in a metric space {{math|(M, d)}} is a continuous function gamma:[0,T] to M. The length of {{math|γ}} is measured byL(gamma)=sup_{0=x_0

• Given a metric space {{math|(X, d)}} and an increasing concave function