Metric Space

Definition

A Metric on a set $X$ is a function $d:X\times X\to [0,\infty ⟩$ such that

1. $d(x,y)=d(y,x),\forall x,y\in X$
2. $d(x,y)=0⟺x=y$
3. $d(x,z)\le d(x,y)+d(y,z)$ (think of this as a triangle and Pythagoras’ Theorem)
   Think of $d(x,y)$ as the distance between $x$ and $y$.