The definition of a metric space

A metric on a set $X$ is a real-valued function $d$ on $X \times X$ that has the following properties:

1. $d(x, y) \geq 0$, $x, y \in X$,
2. $d(x, y) = 0$ if and only if $x = y$,
3. $d (x, y) = d(y, x)$, $x, y in X$,
4. $d (x, z) \leq d (x, y) + d (y, z)$, $x, y, z \in X$. (triangle inequality)

A metric space $(X, d)$ is a set $X$ equipped with a metric $d$ on $X$.