If A1,A2,......An, is a collection of n number of sets in a metric space, prove that their intersection is open.
As written that statement is false.
If each of the $\displaystyle A_i$ is closed and $\displaystyle \bigcap\limits_n {{A_n}} \ne \emptyset $ then $\displaystyle \bigcap\limits_n {{A_n}} $ is closed.
Did you mean each of the $\displaystyle A_i$ is open?