Let $(X,d_X)$ and $(Y,d_Y)$ be metric spaces, and let $1 \le p \le \infty$. For $1 \le p < \infty$, define $d_p:(X\times Y)\times(X\times Y)\to[0,\infty)$ by
Then this [product metric](/page/Product%20Metric) induces on $X \times Y$ exactly the [product topology](/page/Product%20Topology) determined by the metric topology induced by $d_X$ on $X$ and the metric topology induced by $d_Y$ on $Y$.