Metric Space

A set X together with a function d: X × X → [0, ∞) (the metric) satisfying for all x,y,z: d(x,y) ≥ 0 with equality iff x = y (identity of indiscernibles), d(x,y) = d(y,x) (symmetry), and d(x,z) ≤ d(x,y) + d(y,z) (triangle inequality).

A set X equipped with a distance function d: X×X → [0,∞) such that for all x,y,z in X: d(x,y)=0 iff x=y (identity of indiscernibles), d(x,y)=d(y,x) (symmetry), and d(x,z) ≤ d(x,y)+d(y,z) (triangle inequality).

A set X equipped with a function d: X×X → [0,∞) (called a metric or distance) that assigns a nonnegative real number to each ordered pair of points and satisfies: (i) d(x,y)=0 iff x=y (positivity/identity of indiscernibles), (ii) d(x,y)=d(y,x) (symmetry), and (iii) d(x,z) ≤ d(x,y)+d(y,z) (triangle inequality).