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In mathematics, a metric space is a set where a notion of distance (called a metric) between elements of the set is defined. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and Distance is a numerical description of how far apart objects are In Mathematics, a metric or distance function is a function which defines a Distance between elements of a set.

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. In Mathematics, the Euclidean distance or Euclidean metric is the "ordinary" Distance between two points that one would measure with a ruler 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. Geometry ( Greek γεωμετρία; geo = earth metria = measure is a part of Mathematics concerned with questions of size shape and relative position In mathematics non-Euclidean geometry describes how this all works--> hyperbolic and Elliptic geometry, which are contrasted with Euclidean geometry General relativity or the general theory of relativity is the geometric theory of Gravitation published by Albert Einstein in 1916

A metric space also induces topological properties like open and closed sets which leads to the study of even more abstract topological spaces. In Topology and related areas of Mathematics a topological property or topological invariant is a property of a Topological space which is In Metric topology and related fields of Mathematics, a set U is called open if intuitively speaking starting from any point x in In Topology and related branches of Mathematics, a closed set is a set whose complement is open. Topological spaces are mathematical structures that allow the formal definition of concepts such as Convergence, connectedness, and continuity.

Contents

History

Maurice Fréchet introduced metric spaces in his work Sur quelques points du calcul fonctionnel, Rendic. Maurice Fréchet ( September 2, 1878 – June 4, 1973) was a French Mathematician. Circ. Mat. Palermo 22 (1906) 1–74.

Definition

A metric space is an ordered pair (M,d) where M is a set and d is a metric on M, that is, a function

d : M \times M \rightarrow \mathbb{R}

such that

  1. d(x, y) ≥ 0     (non-negativity)
  2. d(x, y) = 0   if and only if   x = y     (identity of indiscernibles)
  3. d(x, y) = d(y, x)     (symmetry)
  4. d(x, z) ≤ d(x, y) + d(y, z)     (triangle inequality). In Mathematics, an ordered pair is a collection of two distinguishable objects one of which is identified as the first coordinate (or the first entry In Mathematics, a metric or distance function is a function which defines a Distance between elements of a set. ↔ In Mathematics, the triangle inequality states that for any Triangle, the length of a given side must be less than or equal to the sum of the other two sides but greater

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. Relaxing the second requirement, or removing the third or fourth, leads to the concepts of a pseudometric space, a quasimetric space, or a semimetric space. In Mathematics, a pseudometric space is a generalized Metric space in which the distance between two distinct points can be zero In Mathematics, a quasimetric space is a generalized Metric space in which the metric is not necessarily symmetric In Topology, a semimetric space is a generalized Metric space in which the Triangle inequality is not required

The first of these four conditions actually follows from the other three, since:

2d(x, y) = d(x, y) + d(y, x) ≥ d(x,x) = 0.

It is more correctly a property of a metric space, but one that many texts include in the definition.

Some authors require the set M to be non-empty.

Metric spaces as topological spaces

The treatment of a metric space as a topological space is so consistent that it is almost a part of the definition.

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}. In Mathematics, a ball is the inside of a Sphere; both concepts apply not only in the three-dimensional space but also for lower and higher dimensions and for metric

These open balls generate a topology on M, making it a topological space. Topological spaces are mathematical structures that allow the formal definition of concepts such as Convergence, connectedness, and continuity. Explicitly, a subset of M is called open if it is a union of (finitely or infinitely many) open balls. In Metric topology and related fields of Mathematics, a set U is called open if intuitively speaking starting from any point x in The complement of an open set is called closed. In Topology and related branches of Mathematics, a closed set is a set whose complement is open. 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. In Topology and related areas of Mathematics, a metrizable space is a Topological space that is homeomorphic to a Metric space.

Since metric spaces are topological spaces, one has a notion of continuous function between metric spaces. In Topology and related areas of Mathematics a continuous function is a Morphism between Topological spaces Intuitively this is a function This definition is equivalent to the usual epsilon-delta definition of continuity (which does not refer to the topology), and can also be directly defined using limits of sequences. In Mathematics, a continuous function is a function for which intuitively small changes in the input result in small changes in the output

Complete metric spaces

A metric space X is said to be complete if every Cauchy sequence converges in X. In Mathematics, a Cauchy sequence, named after Augustin Cauchy, is a Sequence whose elements become arbitrarily close to each other as the sequence That is to say: that d(x_n, x_m) \to 0 implies that there is some y with d(x_n, y) \to 0.

Topological properties

Every metric space is:

Other topological properties become equivalent in metric spaces. In Topology, a branch of Mathematics, a first-countable space is a Topological space satisfying the "first Axiom of countability " In Mathematics, a paracompact space is a Topological space in which every Open cover admits an open locally finite refinement. In Topology and related branches of Mathematics, normal spaces, T4 spaces, T5 spaces, and T6 spaces In Topology and related branches of Mathematics, a Hausdorff space, separated space or T2 space is a Topological space In Topology and related areas of Mathematics a topological property or topological invariant is a property of a Topological space which is In particular,

A compact metric space is second countable. In Topology, a second-countable space is a Topological space satisfying the " second Axiom of countability " [1]

Examples of metric spaces

Notions of metric space equivalence

Comparing two metric spaces one can distinguish various degrees of equivalence. To preserve at least the topological structure induced by the metric, these require at least the existence of a continuous function between them (morphism preserving the topology of the metric spaces). In Topology and related areas of Mathematics a continuous function is a Morphism between Topological spaces Intuitively this is a function In Mathematics, a morphism is an abstraction derived from structure-preserving mappings between two Mathematical structures The study of morphisms and

Given two metric spaces (M1, d1) and (M2, d2):

In case of Euclidean space with usual metric the two notions of similarity are equivalent.

Boundedness and compactness

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. Geometry, a diameter of a Circle is any straight Line segment that passes through the center of the circle and whose Endpoints are on the 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. In Topology and related branches of Mathematics, a totally bounded space is a space that can be covered by finitely many Subsets of any 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 (the first example above) under which it is bounded and yet not totally bounded. A useful characterisation of compactness for metric spaces is that a metric space is compact if and only if it is complete and totally bounded. This is known as Heine–Borel theorem. In the Topology of Metric spaces the Heine–Borel theorem, named after Eduard Heine and Émile Borel, states For a Subset Note that compactness depends only on the topology, while boundedness depends on the metric.

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". In Mathematics, an interval is a set of Real numbers with the property that any number that lies between two numbers in the set is also included in the set In Mathematics, the real numbers may be described informally in several different ways 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.

In a metric space, sequential compactness, countable compactness and compactness are all equivalent.

By restricting the metric, any subset of a metric space is a metric space itself (a subspace) with a topology restricted to that set. We call such a subset complete, bounded, totally bounded or compact if it, considered as a metric space, has the corresponding property. Every closed subspace of a complete metric space is complete, and every complete subspace of a metric space is closed. A closed subspace of a metric space, however, need not be complete.

Separation properties and extension of continuous functions

Metric spaces are paracompact[2] Hausdorff spaces[3] and hence normal (indeed they are perfectly normal). In Mathematics, a paracompact space is a Topological space in which every Open cover admits an open locally finite refinement. In Topology and related branches of Mathematics, a Hausdorff space, separated space or T2 space is a Topological space In Topology and related branches of Mathematics, normal spaces, T4 spaces, T5 spaces, and T6 spaces 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). In Mathematics, a partition of unity of a Topological space X is a set of continuous functions \{\rho_i\}_{i\in I} from X In Topology, the Tietze extension theorem states that if X is a Normal topological space and f: A &rarr R 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. In Mathematics, more specifically in Real analysis, Lipschitz continuity, named after Rudolf Lipschitz, is a smoothness condition for functions

Distance between points and sets

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 sethttp://en.wikipedia.org../../../../articles/d/i/s/Distance.html#Distances_between_sets_and_between_a_point_and_a_set. In Topology and related branches of Mathematics, Tychonoff spaces and completely regular spaces are particularly nice kinds of Topological spaces Distance is a numerical description of how far apart objects are 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. In Mathematics the infimum of a Subset of some set is the Greatest element, not necessarily in the subset that is less than or equal to all elements of In Mathematics, the closure of a set S consists of all points which are intuitively "close to 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\mapsto d(x,S) is continuous.

Product metric spaces; normed product metrics

The following construction is useful to remember:

If (M_1,d_1),\ldots,(M_n,d_n) are metric spaces, and N is any norm on Rn, then

\Big(M_1\times \ldots \times M_n, N(d_1,\ldots,d_n)\Big) is a metric space, where the normed product metric is defined by

N(d_1,...,d_n)\Big((x_1,\ldots,x_n),(y_1,\ldots,y_n)\Big) = N\Big(d_1(x_1,y_1),\ldots,d_n(x_n,y_n)\Big) ,

and the induced topology agrees with the product topology. In Topology and related areas of Mathematics, a product space is the Cartesian product of a family of Topological spaces equipped with a natural

Similarly, a countable product of metric spaces can be obtained using the following metric

d(x,y)=\sum_{i=1}^\infty \frac1{2^i}\frac{d_i(x_i,y_i)}{1+d_i(x_i,y_i)}.

Continuity of distance

It is worth noting that in the case of a single space (M,d), the distance map d:M\times M \rightarrow R^+ (from the definition) is uniformly continuous with respect to any normed product metric N(d,d) (and in particular, continuous with respect to the product topology of M\times M). In Mathematics, a metric space is a set where a notion of Distance (called a metric) between elements of the set is defined

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. In Mathematics, an equivalence relation is a Binary relation between two elements of a set which groups them together as being "equivalent" Given two equivalence classes [x] and [y], we define

d'([x],[y]) = \inf\{d(p_1,q_1)+d(p_2,q_2)+...+d(p_{n},q_{n})\}

where the infimum is taken over all finite sequences (p_1, p_2, \dots, p_n) and (q_1, q_2, \dots, q_n) with [p1] = [x], [qn] = [y], [q_i]=[p_{i+1}], i=1,2,\dots n-1. In Mathematics the infimum of a Subset of some set is the Greatest element, not necessarily in the subset that is less than or equal to all elements of In general this will only define a pseudometric, i. In Mathematics, a pseudometric space is a generalized Metric space in which the distance between two distinct points can be zero e. d'([x],[y]) = 0 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. In Topology and related areas of Mathematics, a quotient space (also called an identification space) is intuitively speaking the result of identifying

The quotient metric d is characterized by the following universal property. In various branches of Mathematics, certain constructions are frequently defined or characterised by an abstract property which requires the existence of a unique Morphism If f:(M,d)\longrightarrow(X,\delta) is a metric map between metric spaces (that is, \delta(f(x),f(y))\le d(x,y) for all x, y) satisfying f(x)=f(y) whenever x\sim y, then the induced function \overline{f}:M/\sim\longrightarrow X, given by \overline{f}([x])=f(x), is a metric map \overline{f}:(M/\sim,d')\longrightarrow (X,\delta).

See also

References

  1. ^ http://planetmath.org/encyclopedia/CompactMetricSpacesAreSecondCountable.html
  2. ^ Rudin, Mary Ellen. In the mathematical theory of Metric spaces a metric map or short map is a Continuous function between metric spaces that does not increase any This is a glossary of some terms used in Riemannian geometry and Metric geometry &mdash it doesn't cover the terminology of Differential topology. Topology ( Greek topos, "place" and logos, "study" is the branch of Mathematics that studies the properties of In Mathematics, the triangle inequality states that for any Triangle, the length of a given side must be less than or equal to the sum of the other two sides but greater In Mathematics, more specifically in Real analysis, Lipschitz continuity, named after Rudolf Lipschitz, is a smoothness condition for functions For the Mechanical engineering and Architecture usage see Isometric projection. In Mathematics, a contraction mapping, or contraction, on a Metric space (Md is a function f from M to itself In the mathematical theory of Metric spaces a metric map or short map is a Continuous function between metric spaces that does not increase any The category Met, first considered by Isbell (1964 has Metric spaces as objects and Metric maps or Short maps as Morphisms In Linear algebra, Functional analysis and related areas of Mathematics, a norm is a function that assigns a strictly positive length An understanding of Calculus and Differential equations is necessary for the understanding of Nonrelativistic physics. A new proof that metric spaces are paracompact. Proceedings of the American Mathematical Society, Vol. 20, No. 2. (Feb. , 1969), p. 603.
  3. ^ metric spaces are Hausdorff on PlanetMath

External links

Cut-the-knot is an educational website maintained by Alexander Bogomolny and devoted to popular exposition of a great variety of topics in Mathematics.

Dictionary

metric space

-noun

  1. (mathematics) Any space whose elements are points, and between any two of which a non-negative real number can be defined as the distance between the points; an example is Euclidean space
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