Proof that Intersection and Union of Closed Spaces are Closed
In a metric space
1. Bothand the empty setare closed (and in fact these are also open).
2. The union of any two closed sets, and any finite number of closed sets is closed.
3. The intersection of anyfamily of closed sets is closed.
1.is open sois closed.is op[en sois closed. (bothand are complements of open sets, hence closed).
2. Supposeandare closed sets. Their complementsandare open so thatis open butsois open sois closed.
If all theare closed thenis open and so is
Thenis open andis closed.