I'm working in baby Rudin. I think understand the ideas of interior and limit points, open and closed sets.
We have the theorem that $E$ is closed iff $E^c$ is open.
This gives me some idea about the relationship between a set and its complement. However, I find myself stuck when it comes to determining whether the following statements are true, and, if so, how to prove them:
1. If $E$ is not open then it is closed, and vice-versa.
- If $E$ is not open and $E^c$ is not closed then $E$ is closed and $E^c$ is open.
Is this statement equivalent to saying that a set is closed if its complement is open?
- Proving the following (This one is a question from Rudin):
Let $E^o$ be the set of all interior points of a set $E$. Prove $E^o$ is always open.
(This is my first time using this markup language so if anyone can suggest improvements or good practices I'd be happy to adopt them).