Prove that the intersection of two open sets is open. I am just starting to learn proofs and my question mainly has to do with the structure.
Let A and B be open sets.
If the statement
$A \cap B \neq \emptyset \implies A \cap B$ is open
is true, then its contrapositive
$A \cap B$ not open $\implies A \cap B = \emptyset$
is also true.
However, we know that the empty set is open as well as closed, so the first statement can't be true.
The problem is, I know that this is a true statement. Where am I going wrong?
 A: In logic $P\Longrightarrow Q$ doesn't require $P$ to be true, it is possible that a false statement implies a true statement (actually, you can prove anything starting from a false statement). $P\Longrightarrow Q$ just says that "if $P$ is true, then $Q$ must be true; if $P$ is false, then the statement is vacuously true".
A: On its own, the statement "$A\cap B\neq\emptyset~~\Longrightarrow ~A\cap B$ open" is then just a convoluted way of saying that $A\cap B$ is open. If it's empty, then it's open because the empty set is always open. And if it's not empty, then it's also open according to our statement. Either way, the set is open.
The contrapositive is then also true: If $A\cap B$ is not open, then $A\cap B=\emptyset$. It might sound like the empty set is not open, but that's just because you're probably thinking: "There must be some non-open intersection of open sets, and that intersection is then empty, making the empty set not open". But the first statement in this argument is simply not true: There is no non-open intersection of open sets, so there's also no non-open set that is empty, so the empty set isn't necessarily not open.
