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)