An Etymological Dictionary of Astronomy and Astrophysics

An set of points such that the distance between every pair of points is defined by a → distance function with
the following properties: 1) the distance from the first point to the second equals zero if and only if the points are the same, 2) the distance from the first point to the second equals the distance from the second to the first, and 3) the sum of the distance from the first point to the second and the distance from the second point to a third exceeds or equals the distance from the first to the third.

In mathematical language, the properties, for a nonempty set X, can be expressed as:

1. d(x,y)≥ 0 and d(x,y) = 0 if and only if x = y.
   
2. d(x, y) = d(y,x) for all x, y ∈ X.
   
3. d(x,z)≤ d(x,y) + d(y,z) for all x, y, and z ∈ X. Also called → triangle inequality.

See also: → metric; → space.