# فضاء متري

في الرياضيات ، دالة المسافة distance function أو المترية metric هي دالة رياضية تعرف المسافة بين العناصر ضمن مجموعة ما .

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 metric, the green path has length ${\displaystyle 6{\sqrt {2}}\approx 8.49}$, and is the unique shortest path, whereas the red, yellow, and blue paths still have length 12.

أي مجموعة مزودة بتابع مسافة تدعى فضاءا متريا metric space. هذه المترية أو دالة المسافة هي التي تخلق طوبولوجيا ضمن هذه المجموعة (أي أنها تحول هذه المجموعة إلى فضاء طوبولوجي), لكن العكس غير صحيح فليست كل طوبولوجيا يتم تشكيلها بوساطة مترية .

عندما تكون الطوبولوجيا قابلة للوصف بوساطة متري نقول أن هذا الفضاء قابل للقياس (مقيس) metrisable .

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[1] and analysis on metric spaces.[2]

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.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

## تعريف

### الدافع

A diagram illustrating the great-circle distance (in cyan) and the straight-line distance (in red) between two points P and 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).

### التعريف

المترية على المجموعة X دالة رياضية (تدعى أيضا دالة المسافة)

d : X × XR

(حيث R مجموعة الأعداد الحقيقية). من أجل x, y, z ضمن X, يقتضي هذه الدالة تحقيق الشروط التالية :

1. d(x, y) ≥ 0     ( اللاسلبية )
2. d(x, y) = 0   if and only if   x = y     ()
3. d(x, y) = d(y, x)     (التناظر)
4. d(x, z) ≤ d(x, y) + d(y, z)     (لامساواة المثلث).

### فضاءات الطول

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 supremum.

A curve in a metric space (M, d) is a continuous function ${\displaystyle \gamma :[0,T]\to M}$ . The length of γ is measured by

${\displaystyle L(\gamma )=\sup _{0=x_{0}

In general, this supremum may be infinite; a curve of finite length is called rectifiable.[3] Suppose that the length of the curve γ is equal to the distance between its endpoints—that is, it's the shortest possible path between its endpoints. After reparametrization by arc length, γ becomes a geodesic: a curve which is a distance-preserving function.[4] A geodesic is a shortest possible path between any two of its points.[أ]

A geodesic metric space is a metric space which admits a geodesic between any two of its points. The spaces ${\displaystyle (\mathbb {R} ^{2},d_{1})}$  and ${\displaystyle (\mathbb {R} ^{2},d_{2})}$  are both geodesic metric spaces. In ${\displaystyle (\mathbb {R} ^{2},d_{2})}$ , geodesics are unique, but in ${\displaystyle (\mathbb {R} ^{2},d_{1})}$ , there are often infinitely many geodesics between two points, as shown in the figure at the top of the article.

The space M is a length space (or the metric d is intrinsic) if the distance between any two points x and y is the infimum of lengths of paths between them. Unlike in a geodesic metric space, the infimum does not have to be attained. An example of a length space which is not geodesic is the Euclidean plane minus the origin: the points (1, 0) and (-1, 0) can be joined by paths of length arbitrarily close to 2, but not by a path of length 2. An example of a metric space which is not a length space is given by the straight-line metric on the sphere: the straight line between two points through the center of the Earth is shorter than any path along the surface.

Given any metric space (M, d), one can define a new, intrinsic distance function dintrinsic on M by setting the distance between points x and y to be infimum of the d-lengths of paths between them. For instance, if d is the straight-line distance on the sphere, then dintrinsic is the great-circle distance. However, in some cases dintrinsic may have infinite values. For example, if M is the Koch snowflake with the subspace metric d induced from ${\displaystyle \mathbb {R} ^{2}}$ , then the resulting intrinsic distance is infinite for any pair of distinct points.

### طيات ريمان

A Riemannian manifold is a space equipped with a Riemannian metric tensor, which determines lengths of tangent vectors at every point. This can be thought of defining a notion of distance infinitesimally. In particular, a differentiable path ${\displaystyle \gamma :[0,T]\to M}$  in a Riemannian manifold M has length defined as the integral of the length of the tangent vector to the path:

${\displaystyle L(\gamma )=\int _{0}^{T}|{\dot {\gamma }}(t)|dt.}$

On a connected Riemannian manifold, one then defines the distance between two points as the infimum of lengths of smooth paths between them. This construction generalizes to other kinds of infinitesimal metrics on manifolds, such as sub-Riemannian and Finsler metrics.

The Riemannian metric is uniquely determined by the distance function; this means that in principle, all information about a Riemannian manifold can be recovered from its distance function. One direction in metric geometry is finding purely metric ("synthetic") formulations of properties of Riemannian manifolds. For example, a Riemannian manifold is a CAT(k) space (a synthetic condition which depends purely on the metric) if and only if its sectional curvature is bounded above by k.[7] Thus CAT(k) spaces generalize upper curvature bounds to general metric spaces.

## التاريخ

In 1906 Maurice Fréchet introduced metric spaces in his work Sur quelques points du calcul fonctionnel[8] 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.[9]

General metric spaces have become a foundational part of the mathematical curriculum.[10] 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.[11] 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.

## ملاحظات

1. ^ This differs from usage in Riemannian geometry, where geodesics are only locally shortest paths. Some authors define geodesics in metric spaces in the same way.[5][6]

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

## الهامش

1. ^
2. ^
3. ^ Burago, Burago & Ivanov 2001, Definition 2.3.1.
4. ^
5. ^ Burago, Burago & Ivanov 2001, Definition 2.5.27.
6. ^ Gromov 2007, Definition 1.9.
7. ^ Burago, Burago & Ivanov 2001, p. 127.
8. ^ Fréchet, M. (December 1906). "Sur quelques points du calcul fonctionnel". Rendiconti del Circolo Matematico di Palermo. 22 (1): 1–72. doi:10.1007/BF03018603. S2CID 123251660.
9. ^ Blumberg, Henry (1927). "Hausdorff's Grundzüge der Mengenlehre". Bulletin of the American Mathematical Society. 6: 778–781. doi:10.1090/S0002-9904-1920-03378-1.
10. ^ Rudin 1976, p. 30.
11. ^ E.g. 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".

## المراجع

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .