Product metric

Definition

Let $(X, d_X)$ and $(Y, d_Y)$ be metric spaces. The product metric on the Cartesian product $X \times Y$ is the function $d$ with $$d\bigl((x_1, y_1), (x_2, y_2)\bigr) = \max\bigl(d_X(x_1, x_2), \, d_Y(y_1, y_2)\bigr)$$ for all $x_1, x_2 \in X$ and $y_1, y_2 \in Y$.

Remarks

The function $d$ is a metric on $X \times Y$. Other common product metrics are $d_X + d_Y$ and $\sqrt{d_X^2 + d_Y^2}$. Each of them is between $d$ and $2d$, so all of them are equivalent and give the product topology. For $X = Y = \mathbb{R}$ with the usual metric, $d$ is the sup metric on $\mathbb{R}^2$. The same formula with a maximum over $n$ factors gives a product metric on a product of $n$ metric spaces.

Explanations

L1 human

Two pairs are close in the product metric when their first parts are close and their second parts are close. The maximum is the worse of the two distances.