Question: Let $X$ be any set with at least two elements. Assume that the only open subsets of $X$ are the empty set $\emptyset$ and $X$ itself. - Which subsets of $X$ are closed? - Which subsets of $X$ are compact?
My thoughts: Thus also $\emptyset$ and $X$ have to be also closed subsets. As their complements are both open and by definition the set is closed if the complement is open.
The open set is compact as it is a finite set and also $X$ is compact as it has a finite amount of closed subsets, thus is bounded and closed. Am I in any way correct with these thoughts?