Metric space

Definition

Let $X$ be a set. A metric on $X$ is a function $d \colon X \times X \to \mathbb{R}$ with these properties, for all $x, y, z \in X$:

1. $d(x, x) = 0$.
2. If $x \neq y$, then $d(x, y) > 0$.
3. $d(x, y) = d(y, x)$.
4. $d(x, z) \le d(x, y) + d(y, z)$.

A metric space is a pair $(X, d)$, where $X$ is a set and $d$ is a metric on $X$. The number $d(x, y)$ is the distance between $x$ and $y$.