metric space

In mathematics, a metric space is a set where a notion of distance (called a metric) between elements of the set is defined. 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. The Euclidean metric defines the distance between two points as the length of the straight line connecting them.
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Noun
1. a set of points such that for every pair of points there is a nonnegative real number called their distance that is symmetric and satisfies the triangle inequality
(hypernym) mathematical space, topological space
(hyponym) Euclidean 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) 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) 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)