Canonical metric on product of two complete metric spaces

Question

Suppose we have two metric spaces $(X, d_X)$ and $(Y, d_Y)$ and consider product of sets $X\times Y$. There is well known statement: ''Product of two complete metric spaces is complete''. Thinking about this question i came up with the following: i can show for a lot of metrics on $X \times Y$ that it's true and i can construct a lot of metrics, but is there any canonical metric on $X \times Y$?I mean, the product must be complete with respect to some metric, so it's true for an arbitrary metric on product?Is there any connection with product topology?

Answer 1

You can define, for instance,$$d\bigl((x_1,y_1),(x_2,y_2)\bigr)=\max\bigl\{d_X(x_1,x_2),d_Y(y_1,y_2)\bigr\}$$or$$d\bigl((x_1,y_1),(x_2,y_2)\bigr)=d_X(x_1,x_2)+d_Y(y_1,y_2),$$or even$$d\bigl((x_1,y_1),(x_2,y_2)\bigr)=\sqrt{d_X(x_1,x_2)^2+d_Y(y_1,y_2)^2}.$$Any of them will do. And they all induce the product topology.

But it is not true that $X\times Y$ is complete with respect to any distance that you define on it, even if it induces the product topology.