Equivalence of Definitions of Topology Induced by Metric

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

$(1)$ implies $(2)$
Let $T = \left({A, \tau_d}\right)$ be the topological space of which $\tau_d$ is the topology induced on $M$ by $d$ by definition 1.

Then by definition:
 * $\tau_d$ is the set of all open sets of $M$.

...

Thus $\tau_d$ is the topology induced on $M$ by $d$ by definition 2.

$(2)$ implies $(1)$
Let $T = \left({A, \tau_d}\right)$ be the topological space of which $\tau_d$ is the topology induced on $M$ by $d$ by definition 2.

Then by definition:
 * $\tau$ is the topology generated by the basis consisting of the set of all open $\epsilon$-balls in $M$.

...

Thus $\tau_d$ is the topology induced on $M$ by $d$ by definition 1.