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

,

and

such that

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?