Lecture Notes

(a) Only proving the statement in one direction

(b) $E$ is not closed doesn't mean $E$ is open. There is no dichotomy between open and closed sets. Some sets maybe both open and closed, like $\emptyset$, some sets maybe neither open and closed, like $[0,1)$ in $\R$. And the notion of open and closed is a property of subset, so we need to say the full sentence like, 'the subset $A \In S$ is closed in $S$'. Say a set is closed or open without saying what's the ambient space has no meaning.