Given the proposition: A subset U of a metric space is an open set iff U is a union of open balls. Then we are told that it follows from the above proposition that two metrics are equivalent if each open ball of either of the metric topolgies is an open set of the other topology. This is really confusing. Firstly if a metric topology is essentially a set of sets without a materic how do we define an open ball in it? Does a metric topology somehow inherit a metric from it's corresponding metric space? Any thoughts or points that might help me undestand this would be appreciated thanks.