O que (quem) é metric space - definição

SET EQUIPPED WITH A METRIC (DISTANCE FUNCTION)

Distance function; Metric spaces; Metric topology; Metric geometry; Bounded metric space; Bounded metric; Bounded space; Metric (mathematics); Quasimetric space; Translation-invariant metric; Translation invariant metric; Semi-metric; Semi metric space; Homogeneous metric; Quotient metric space; Semimetric; Postoffice metric; Post office metric; British rail metric; SNCF metric; Quasimetric; Quasi-metric; Premetric space; Prametric; Hemimetric space; Hemimetric; Semimetric space; Hemi-metric; Prametrizable; Distance to a set; Metric distance; General metric; Relation of norms and metrics; Inframetric; Post Office metric; Prametric space; Premetric; Finite metric space; Discrete metric space; Metric Geometry; Distance metric; Pseudoquasimetric space; Pseudo-semimetric; French railway metric; British Rail metric; Norm induced metric; Diameter of a metric space; Generalizations of metric spaces; Metric embeddings

$length]] (12), and are all shortest paths. In the Euclidean metric, the green path has length <math>6 \sqrt{2} \approx 8.49</math>, and is the unique shortest path, whereas the red, yellow, and blue paths still have length 12.$

metric space

<mathematics> A set of points together with a function, d, called a metric function or distance function. The function assigns a positive real number to each pair of points, called the distance between them, such that: 1. For any point x, d(x,x)=0; 2. For any two distinct points x and y, d(x,y) > 0; 3. For any two points x and y, not necessarily distinct, d(x,y) = d(y,x). 4. For any three points x, y, and z, that are not necessarily distinct, d(x,z) <= d(x,y) + d(y,z). The distance from x to z does not exceed the sum of the distances from x to y and from y to z. The sum of the lengths of two sides of a triangle is equal to or exceeds the length of the third side. (2003-06-26)

Metric space

In 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.

https://en.wikipedia.org/wiki/Metric_space

Metric (mathematics)

In mathematics, a metric or distance function is a function that gives a distance between each pair of point elements of a set. A set with a metric is called a metric space.

https://en.wikipedia.org/wiki/Metric_(mathematics)

Wikipédia

Metric space

In 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. 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 and analysis on metric spaces.

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.

https://en.wikipedia.org/wiki/Metric space