Let $X$ be a set, and let $d:X\times X\to\mathbb{R}$ be the [discrete metric](/page/Discrete%20Metric), so that $d(x,y)=0$ if $x=y$ and $d(x,y)=1$ if $x\neq y$. If $\tau_d$ denotes the metric topology induced by $d$, then $\tau_d=\mathcal{P}(X)$. Equivalently, the metric topology induced by the discrete metric is the discrete topology on $X$.