Hello, This is my problem: let be a distance on X. We know d is continous with the prodcut topology when the topology given to X is the metric topology. Show tha if d is continous with the product topology when the topology given to X is then this topology is finner than the metric toplogy.

Attempts: we know that there is a coarsest topology for which d is continous is open set in R I tried to prove that this topology is the product topology of the metric but this doesn't seem to be the case.

I also tried to prove directly that if U is open in the metric topology then U is in

Any comments will be appreciated.