Closure (topology) - Facts About Closures

Facts About Closures

The set is closed if and only if . In particular, the closure of the empty set is the empty set, and the closure of itself is . The closure of an intersection of sets is always a subset of (but need not be equal to) the intersection of the closures of the sets. In a union of finitely many sets, the closure of the union and the union of the closures are equal; the union of zero sets is the empty set, and so this statement contains the earlier statement about the closure of the empty set as a special case. The closure of the union of infinitely many sets need not equal the union of the closures, but it is always a superset of the union of the closures.

If is a subspace of containing, then the closure of computed in is equal to the intersection of and the closure of computed in : . In particular, is dense in if and only if is a subset of .