A theorem in my book says:
Suppose that are metric spaces. Define a function . Then f is continuous iff is open in for every open set .
Proof in the book goes:
Assume that for each open , is open in .
Let , then .
Then , is open in since is open in , therefore f is continuous.
Q1: I understand why the set in the inverse is open, but how does that implies f is continuous? In other words, if I fill in the details, say:
Let , then , pick , then whenever , I need to get But how did the statement from above help me to get there?
Conversely, suppose that f is continuous, let U be open set in .
, there exists ,
Implies that is open.
Q2: I understand that the open ball around y is in U since it is an open set, why I'm a bit unsure about how to get , I'm sure it is because f is continuous. I know that continuous function map an open set to an open set, but do I know for sure it is true for the inverse?