Source : WordNet®

metric space
     n : 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

Source : Free On-Line Dictionary of Computing

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)