Discrete Metric induces the Discrete Topology

Last updated Nov 1, 2022

Let $S$ be a Set equipped with the Discrete Metric, $d$. Then the Metric Topology induced by $d$ is the Discrete Topology on $S$.

Observe that $B_{1}(x) = {x}$ is a Open in the Metric Topology for all $x \in X$. Thus, the Metric Topology contains all the singletons. Then the Metric Topology contains the Discrete Topology. Since the Discrete Topology is the finest possible Topological Space, the Metric Topology here must be the Discrete Topology. $\blacksquare$