Standard Discrete Metric induces Discrete Topology

Theorem
Let $M = \struct {A, d}$ be the (standard) discrete metric space on $A$.

Then $d$ induces the discrete topology on $A$.

Thus the discrete topology is metrizable.

Proof
Let $a \in A$.

From Subset of Standard Discrete Metric Space is Open, a set $U \subseteq A$ is open in $M$.

So, in particular, $\set a$ is open in $\struct {A, d}$.

This holds for all $a \in A$.

From Metric Induces Topology it follows that $\set a$ is an open set in $\struct {A, \tau_{A, d} }$.

The result follows from Basis for Discrete Topology.