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 exists $r>0$ with $B_d(x,r)\subseteq U$, where $B_d(x,r)$ is the open ball of radius $r$ centered at $x$.

This makes $(X,\tau_d)$ a topological space; equivalently, the family of open balls forms a basis for $\tau_d$, and the open sets are precisely unions of open balls.