general topology - True or false: sets, subsets, and topologies in $mathbb R$

I am pondering the following statements about sets, subsets and topologies in $\mathbb R$.

The empty set is a closed subset of $\mathbb R$ regardless of the topology on $\mathbb R$.

Any open interval is an open subset of $\mathbb R$ regardless of the topology on $\mathbb R$.

Any closed interval is a closed subset of $\mathbb R$ regardless of the topology on $\mathbb R$.

A half-open interval of the form $[a, b)$ is neither an open set nor a closed set regardless of the topology on $\mathbb R$. I think this is a false statement but I am unsure about the first 3. I am in an introduction to proofs class and we are touching on topology. I know these are important distinctions to make because my professor keeps commenting how their is still a lot of confusion about these statements.