Definition:Topology Induced by Metric

Definition 1
Let $M = \left({A, d}\right)$ be a metric space.

The topology (on $A$) induced by (the metric) $d$ is defined as the set $\tau$ of all open sets of $M$.

Definition 2
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

 * Metric Induces Topology, in which it is shown that $\tau$ is, in fact, a topology on $M$
 * Subspace Topology (sometimes also called the induced topology)