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Definition. A metric space is a set X together with a function d (called a metric or distance function) which assigns a real number d (x, y) to every pair x, y X satisfying the properties (or axioms): 1  d (x, y) 0 and d (x, y) = 0 x = y, 2  d (x, y) = d (y, x), 3  d (x, y) + d (y, z) d (x, z).
In an affine space, there is no distinguished point that serves as an origin. Hence, no vector has a fixed origin and no vector can be uniquely associated to a point. In an affine space, there are instead displacement vectors, also called translation vectors or simply translations, between two points of the space.