I have some problems with the arbitrary union and finite intersections of sets.
Let C be the set of finite unions of subsets of $\mathbb{N}$ of the form $$ \left( n \right) = \left\{ {n,2n,3n,...} \right\} $$ and the empty set.
The problem is to show that the sets of this form are the closed sets of a topology on the set of natural numbers.