It is not a metric space, for two distinct events, a photon leaving the sun eight minutes ago and its.
It is reified, locally, in those regions of a manifold where there are valid metric space type representations.
metric space in Technology
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)