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.

Examples:

• On $\mathbb{R}^n$ with the Euclidean metric, $\tau_d$ is the usual topology of Euclidean space.
• On a set $X$ with the discrete metric, $\tau_d$ is the discrete topology (every subset is open).