You can define closed sets positively as well. A closed set is one which includes its whole boundary. An open set is one which excludes its whole boundary. Clea

You can define closed sets positively as well. A closed set is one which includes its whole boundary. An open set is one which excludes its whole boundary. Clearly then it is possible for a set to contain only part of its boundary, in which case it's neither closed nor open. Somewhat less obviously, it's possible for a set's boundary to be empty, in which case it is both closed and open.