Problem: Prove that the union of any collection of open sets is open.
Is the following proof acceptable?
Proof: Let U' denote the collection of open sets and let U' = , where n = 1,...,k. Since U' is a collection of open sets, there exists an interior point that belongs to U'. Since belongs to U', it is implied that belongs to at least one . So for that belongs to , there exists a neighborhood in ; For that belongs to , there exists a neighborhood in ,...If we take the union of all , there exists a neighborhood in U' for all . Therefore, U' is open.