Metric Space

A set and a function which defines the distance between two points on that set

An ordered pair (X,d), where X is an arbitrary set and d is a metric (distance-defining function).

The metric defines the following properties:

1. d(a,b)=0 iff a=b
2. d(a,b)>0 (if a≠b)
3. d(a,b)=d(b,a)
4. d(a,c)≤d(a,b)+d(b,c)

(Note: #4 is a restatement of the Triangle Inequality)

Give the above axioms, an additional, implicit axiom is

d(a,b)≥0

which stands in agreement with an intuitive formation of the concept of distance.