WordNet (r) 3.0 (2006):

metric space
    n 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

The Free On-line Dictionary of Computing (30 December 2018):

metric space

    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)