Definition (Metric Space)

A metric space is a pair $(X,d)$ where $X$ is a set and $d:X\times X\to \mathbb{R}_{\ge0 }$ s.t. $d$ satisfies an equivalence relation:

1. Reflexive: $d(x,y)=0\iff x=y$
2. Symmetric: $d(x,y)=d(y,x)$
3. Transitive: $d(x,z)\le d(x,y)+d(y,z)$ where $x,y,z\in X$.

We refer to $d$ as the Metric or distance.