The open set criterion
I'm starting to study differential geometry and my starting point is the book "Introduction to Topological Manifolds". It seems to be an appropriate choice but it does not present any solutions so, if you are willing, I ask you to review my solution to Exercise A.16.
Exercise: Prove that
"A map between metric spaces is continuous if and only if the inverse image of every open set is open."
We start off with the definitions.
- Given any set and a distance function satisfying the symmetry, positivity and the triangle inequality properties we say that the pair defines a metric space.
- A function is continuous if and only if
- A set is open if and only if
- Given a function , the inverse image of a set is given by
Let us first prove that if is continuous and is open then is also open, i.e.
For any , there exists such that because we assumed to be open. From the continuity of we conclude that there exists such that . From the definition of , we have that . Therefore, for we have that .
Now, we prove that if is open and is also open then is continuous. Suppose that it is not, then the following holds
Therefore, for each such that we have that , because is an open set. From the definition of we conclude that there exists . Take y=x, then and because of the properties of the distance function, but this contradicts and thus must be continuous.