# Definition
A [[Metric]] on a set $X$ is a function $d : X \times X \to [ \, 0, \infty \rangle$ such that
1. $d(x,y) = d(y, x), \; \forall x,y \in X$
2. $d(x, y) = 0 \iff x = y$
3. $d(x,z) \leq d(x,y) + d(y, z)$ (think of this as a triangle and Pythagoras' Theorem)
Think of $d(x, y)$ as the distance between $x$ and $y$.