Derivation of metric from product topology?

Question

Suppose you have two topological spaces, g11 and g22, which are "components" of a more general topology. For example, suppose that a metric has components g11, g12, g21, and g22. And suppose you want to find a new metric which describes the g11 × g22 topology (the Cartesian product of g11 and g12). How would you find a metric that describes the new g11 × g22 topology?

Answer 1

The product metric construction is the standard method of putting a metric on a product of metric spaces.

If $A$ and $B$ are not equipped with a metric to start with, you would need to verify metrizability first. The Nagata-Smirnov metrization theorem gives necessary and sufficient conditions for metrizability.