- Related Topics:
- topological space
metric space
-
What is a metric space in mathematics?
In mathematics, particularly in topology, a metric space is an abstract set with a distance function (or metric). For any pair of its points, the metric specifies a non-negative distance, holding these properties:
- The distance from a first point to a second is zero if the points are identical.
- The distance from a first point to a second equals the distance from the second to the first.
- The sum of the distance from a first point to a second and the distance from the second point to a third is greater than or equal to the distance from the first to the third. This is the triangle inequality.
The French mathematician Maurice Fréchet started studying metric spaces in 1905.
4 Britannica SourcesCreated from Britannica articles using AI. AI can make mistakes, so verify using Britannica articles. -
What is a metric, and how is it used in a metric space?
In mathematics, particularly topology, a metric space is an abstract set with a distance function. This function, called a metric, assigns a non-negative distance between any two points in the set, adhering to specific properties.
These properties include:
* The distance from a point to itself is zero.
* The distance between two points is the same regardless of the order.
* The sum of the distances from one point to a second and from the second point to a third is greater than or equal to the distance from the first to the third (the triangle inequality).A metric generalizes the notion of usual distance to more general settings.
2 Britannica SourcesCreated from Britannica articles using AI. AI can make mistakes, so verify using Britannica articles. -
What are some common examples of metric spaces?
In mathematics, particularly in topology, a metric space is an abstract set with a distance function (metric) that indicates a non-negative distance between any two points, adhering to specific properties.
Common examples of metric spaces include:
* The real number line with the usual distance function.
* Euclidean n-dimensional space with the usual distance function.
* A set of points with the discrete metric, where the distance from a point to itself is 0, and the distance between distinct points is 1.
* The Euclidean plane with the taxicab metric, where the distance from a point (x, y) to a point (z, w) is |x − z| + |y − w|.
* Sets of bounded real-valued continuous or integrable functions in analysis, which have several useful metrics.2 Britannica SourcesCreated from Britannica articles using AI. AI can make mistakes, so verify using Britannica articles. -
What rules must a metric follow to be valid in a metric space?
In mathematics, particularly topology, a metric space is an abstract set with a distance function (metric) that defines a non-negative distance between any two points. For a metric to be valid, it must satisfy these rules:
- The distance from a point to itself is zero, and the distance between two distinct points is always positive.
- The distance from point A to point B is the same as the distance from point B to point A. In other words, the metric must be symmetric.
- The metric must satisfy the triangle inequality: The distance from point A to point B, plus the distance from point B to point C, must be greater than or equal to the distance from point A to point C.
3 Britannica SourcesCreated from Britannica articles using AI. AI can make mistakes, so verify using Britannica articles. -
How does the concept of distance work in different metric spaces?
In different metric spaces, the concept of distance is defined by a metric, which is a function that assigns a non-negative distance between any two points while adhering to specific properties.
These properties include:
* The distance from a point to itself is zero, and the distance between distinct points is positive.
* The distance from point A to point B is the same as the distance from point B to point A.
* The triangle inequality holds, meaning the distance from point A to point B plus the distance from point B to point C is greater than or equal to the distance from point A to point C.Examples of distance in different metric spaces:
* Euclidean distance In Euclidean space, the Euclidean distance is the length of a straight line segment connecting two points, based on the Pythagorean theorem.
* Taxicab metric The taxicab metric calculates distance as the sum of the absolute differences of their coordinates.
* Discrete metric The discrete metric defines the distance between a point and itself as 0, and the distance between any two distinct points as 1.6 Britannica SourcesCreated from Britannica articles using AI. AI can make mistakes, so verify using Britannica articles. -
Why are metric spaces important in mathematics and what are they used for?
In mathematics, a metric space is an abstract set with a distance function (or metric) that defines a non-negative distance between any two points, following specific properties. These include the distance from a point to itself being zero, the distance between two points being the same in either direction, and the triangle inequality, where the sum of distances between three points is always greater than or equal to the direct distance between the first and third points.
Metric spaces are important because they generalize the idea of distance, which allows mathematicians to define open sets and create examples of topological spaces. They are also used in analysis to define metrics on sets of continuous or integrable functions.
2 Britannica SourcesCreated from Britannica articles using AI. AI can make mistakes, so verify using Britannica articles.
metric space, in mathematics, especially topology, an abstract set with a distance function, called a metric, that specifies a nonnegative distance between any two of its points in such a way that the following properties hold: (1) the distance from the first point to the second equals zero if and only if the points are the same, (2) the distance from the first point to the second equals the distance from the second to the first, and (3) the sum of the distance from the first point to the second and the distance from the second point to a third exceeds or equals the distance from the first to the third. The last of these properties is called the triangle inequality. The French mathematician Maurice Fréchet initiated the study of metric spaces in 1905.
The usual distance function on the real number line is a metric, as is the usual distance function in Euclidean n-dimensional space. There are also more exotic examples of interest to mathematicians. Given any set of points, the discrete metric specifies that the distance from a point to itself equal 0 while the distance between any two distinct points equal 1. The so-called taxicab metric on the Euclidean plane declares the distance from a point (x, y) to a point (z, w) to be |x − z| + |y − w|. This “taxicab distance” gives the minimum length of a path from (x, y) to (z, w) constructed from horizontal and vertical line segments. In analysis there are several useful metrics on sets of bounded real-valued continuous or integrable functions.
Thus, a metric generalizes the notion of usual distance to more general settings. Moreover, a metric on a set X determines a collection of open sets, or topology, on X when a subset U of X is declared to be open if and only if for each point p of X there is a positive (possibly very small) distance r such that the set of all points of X of distance less than r from p is completely contained in U. In this way metric spaces provide important examples of topological spaces.
" height="20" alt="HEW Logo">
A metric space is said to be complete if every sequence of points in which the terms are eventually pairwise arbitrarily close to each other (a so-called Cauchy sequence) converges to a point in the metric space. The usual metric on the rational numbers is not complete since some Cauchy sequences of rational numbers do not converge to rational numbers. For example, the rational number sequence 3, 3.1, 3.14, 3.141, 3.1415, 3.14159, … converges to π, which is not a rational number. However, the usual metric on the real numbers is complete, and, moreover, every real number is the limit of a Cauchy sequence of rational numbers. In this sense, the real numbers form the completion of the rational numbers. The proof of this fact, given in 1914 by the German mathematician Felix Hausdorff, can be generalized to demonstrate that every metric space has such a completion.