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.