It's not a common false belief, IMO. On the first day you learn about open and closed sets, you learn that the for any metric space S, the set S is open in S and the empty set is open in S, and they're closed as well. (Students typically get introduced the concepts in the context of metric spaces.)

Note that sets are always closed or open relative to some other specified (often implied) set S.

In some courses, a closed set (in S) is defined to be a set which is the complement S-T for some open set T. In others, a closed set (in S) is defined to be a set which contains all its limit points (in S). And then whichever isn't the definition gets proven as a theorem.