" The intersection of every sequence of open subsets of R is a borel set.However set of all such intersections is not the borel set because it is not closed under countable unions"
I cannot see why the set is not closed under countable unions. Any countable union of such sets can be expressed as the countable intersection of opensets using the fact that countable union of countable sets is countable. More specifically if the first set is the intersection of all O_1n Second set O_2n and so on... Then union of all such sets can be written as intersection of the sets where each set is the union of a possible combination of O _1(n_1),O_2(n_2) and so on...certainly number of such possibilities is countable. I dont understand where i'm going wrong.