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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).
a three dimensional euclidean space is a special case of a euclidean space in terms of bourbaki the species of three dimensional euclidean space is richer than the species of euclidean space likewise the species of compact topological space is richer than the species of topological space