Is there a natural topology on the set of open sets?

So, if there is an infinite set I and it is described as a set of sets, what can we say about the nature of all element sets--they all can be open or all have to be closed or some have to be both?

• please think on the lines of function f to be defined on each set such that it is complex valued and has k continuous differentials.

• Answer:

  You can't really say, The topology is independent from the base set. I could say only the whole set and empty set are open, or I could say all subsets are open, (thus all sets are closed as well.) Reading again it seems you're asking about the nature of the singletons. You could set it up so all of them are open (a T_1 space) Or none of them open nor closed. This is again because the topology is independent in relation to the base set. (This should be qualified, you can't for example have the topology of the reals on a countable set, so there is some relation to the topology and base set, but none that lets you say anything about the topology and the base set.)