Let $(X,\tau)$ be a [topological space](/page/Topological%20Space) with the discrete topology, so $\tau=\mathcal{P}(X)$. Define the [discrete metric](/page/Discrete%20Metric) $d:X\times X\to\mathbb{R}$ by $d(x,y)=0$ if $x=y$ and $d(x,y)=1$ if $x\neq y$. Then $d$ is a metric on $X$ and the metric topology $\tau_d$ induced by $d$ satisfies $\tau_d=\tau$. In particular, $(X,\tau)$ is metrizable.