Metric spaces

Definition. A metric on $X$ is a map $d:X\times X\to[0,+\infty[$ such that for all $x,y,z\in X$:
1. $d(x,y)=0\iff x=y$;
2. $d(x,y)=d(y,x)$;
3. $d(x,z)\le d(x,y)+d(y,z)$.

Examples.
- Euclidean metric on $\mathbb{R}^n$: $d(x,y)=\|x-y\|_2$.
- Discrete metric: $d(x,y)=0$ if $x=y$, $d(x,y)=1$ otherwise.

Conclusion. A metric space is a pair $(X,d)$; the metric encodes distance and topology.