Metric Spaces

1.    Between any pair of points there is a pathway, and any pathway can be measured by a distance: a non-negative real number.

d(A,B) ∈ [0,∞)

2.    The distance between two points is zero precisely when they are the same point.

d(A,B)=0 iff A=B

3.    Distance is independent of the direction of travel: for any two points A and B, the distance from A to B is the same as from A to B.

d(A,B)=d(B,A)

4.    In going to C from A it is no shorter to go through B than to go directly. This is the familiar “triangle inequality.” For points A, B, and C and distance function d,

d(A,C) ≤ d(A,B) + d(B,C)