You only need to prove that for every there exists an open ball and not necessarily that is the product of two open balls.
I am trying to show that if U is open in a metric space and is open in . Then is open in the product topology generated by the taxicab metric . In order to show this, I tried to prove that .
Let and and
However, there is a problem with the 3rd equivalent inequality since it only holds one direction, not the other. I am not sure if I approach this problem correctly. Hope someone can help me on this.