Definition:Topology Induced by Metric/Definition 2

Definition
Let $M = \left({A, d}\right)$ be a metric space.

The topology (on $A$) induced by (the metric) $d$ is defined as the topology $\tau$ generated by the basis consisting of the set of all open $\epsilon$-balls in $M$.

Also see

 * Equivalence of Definitions of Topology Induced by Metric