In a book on beginning measure theory, the following statement is made: "It is clear that any intersections and finite unions of closed sets are closed." However the intersection of two disjoint closed sets is the empty set which is open by definition. Is there something wrong with my understanding, or was the statement false?
You seem to believe that if a set is open then it is not closed. This is false. In particular, the empty set and the whole space are always both open and closed.