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From The Free On-line Dictionary of Computing (05 January 2017) [foldoc]:
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)