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 $a \neq b$)
3. $d(a, b) = d(b, a)$
4. $d(a, c) \leq d(a, b) + d(b, c)$

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

Give the above axioms, an additional, implicit axiom is

$d(a, b) \geq 0$

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