Let . We say that is an open set if and only if . All of the points in A are interior.
Open does not imply not Closed
Remember that a set being open doesn’t imply it’s not closed. Similarly, a set being closed doesn’t imply it’s not open.
The empty set is an example of a set that is open and closed at the same time.
Union and Intersection
From this theorem, it also follows that if and are closed, then their union and intersection are also closed.