metric space

A metric space is a set together with a notion of distance between its elements, called points, which is measured by a metric.

Formally,
a metric space is an ordered pair $(M,d)$ where $M$ is a set and $d$ is a metric on $M$, i.e. a function $d:M\times M\to \mathbb{R}$ satisfying the metric space axioms for all points $x,y,z\in M$:

1. the distance from a point to itself is zero: $$d(x,x)=0$$
2. (positivity) The distance between two distinct points is always positive: $$x\neq y \implies d(x,y)>0$$
3. (symmetric) The distance from $x$ to $y$ is always the same as the distance from $y$ to $x$: $$d(x,y)=d(y,x)$$
4. The triangle inequality holds: $$d(x,z)\leq d(x,y)+d(y,z)$$