Suppose is a metric space and the corresponding product topology is defined on . Prove that the metric d is continuous.
You have to show that if is close to in , then is close to . More precisely, you have to show that, given and , there exists such that if and then . Use the triangle inequality.
Suppose is a metric space and the corresponding product topology is defined on . Prove that the metric d is continuous.
I'm sorry. What is this question actually asking. I see that Opalg has already answered it, but I am just curious. Is it that if is a metric space then is continuous?
Is it that if is a metric space then is continuous?
Yes, the question is as you say. But the problem involves product topology and I am wondering whether it should necessarily be defined on to prove the statement.