nLab metric topology

Definition

Given a metric space $(X,d)$, the metric topology on $X$ is the structure of a topological space $\mathcal{T}$ on $X$ which is generated from the topological base of $\mathcal{T}$ given by the open balls

$$B(x,r) \coloneqq \{y \in X \;|\; d(x,y) \lt r \}$$

for all $x \in X$ and $r \in (0,\infty) \subset \mathbb{R}$.

A topological space whose topology is the metric topology for some metric space structure on its underlying set is called a metrizable topological space.

Related concepts

• continuous function