Topological spaces and open sets

Question

I would like someone to look through my solution to this problem and let me know if I have it correct or if I need to change anything. Any help would be greatly appreciated!

Answer 1

(a) is directly from the definition you gave. There is nothing to prove.

(b) The example is correct, but the fact that $\{0\}$ is closed does not imply it is not open. (Sets can be open and closed at the same time and not closed does not imply open; sets aren't doors!) For that you have to show that $0$ is not an interior point of $\{0\}$, using whatever definition you have for the topology on$\mathbb{R}$.