Metric-induced topology

The metric-induced topology on a metric space $(X,d)$ is the collection $\tau_d$ of subsets $U\subseteq X$ such that, for every $x\in U$ , there is $r>0$ with $B_d(x,r)\subseteq U$ . Here $B_d(x,r)$ is the open ball of radius $r$ centered at $x$ .

Basis

The family of open balls is a basis for $\tau_d$ , so the open sets are precisely the unions of open balls.

Examples

On $\mathbb R^n$ with the Euclidean metric, $\tau_d$ is the usual Euclidean topology.
On a set $X$ with the discrete metric, $\tau_d$ is the discrete topology.