Metric space

A metric space is a pair $(X,d)$ where $X$ is a set and $d$ is a metric on $X$, i.e. a function $d:X\times X\to[0,\infty)$ satisfying positive definiteness, symmetry, and the triangle inequality.

Metric spaces generalize Euclidean spaces and provide the setting for “analysis without coordinates.” Many results in real analysis extend to general metric spaces once stated in terms of $d$.

Examples:

• $(\mathbb{R},|x-y|)$ is a metric space.
• $(\mathbb{R}^k,\|x-y\|_2)$ is a metric space.
• Any set $X$ with the discrete metric is a metric space in which every subset is open.