I just want to confirm if my proof is correct for the statement: "The empty set is closed". Although I know that the empty set is also open, the proof is similar.
Proof. Suppose ∅ is open, then for all x that is an element of ∅ there exist a ball B(x,r) that is centered at x with radius r such that B(x,r) is a subset of ∅. But this implies that ∅ is not the empty set since it contains some elements. Thus, if ∅ is open then ∅ is not the empty set, but arguing contrapositively, if ∅ is the empty set then ∅ is closed. QED