Metric Space

A metric space $(X, d)$ is a Topological Space with a metric $d(x,y): X \times X \rightarrow \mathbb{R}$ defined on members of the set. This metric is a generalization of distance, with the following properties:

where property $(4)$ is the triangle inequality. Also, the metric generates the topology on the open sets; a basis can be chosen by including every open ball, which is defined as $B(x, r) = \lbrace y: d(x, y) < r\rbrace$. A neighbourhood basis can be chosen by including every open rational ball that is a neighbourhood of $x$, and in fact this neighbourhood basis is countable, so metric spaces are first countable.