Metric space

A metric space is a pair $(X,d)$ where $X$ is a set and $d$ is a metric on $X$.

Every metric space determines a topological space via the metric-induced topology , whose basic neighborhoods are open balls .

Examples:

• $(\mathbb{R},d)$ with $d(x,y)=|x-y|$.
• Any set $X$ with the discrete metric $d(x,y)\in\{0,1\}$.