I assume you mean arbitrary intersections because both "finite unions" and "arbitrary unions" conditions at the same time do not make much sense.

Cofinite topology on R

Finite point sets are closed in the cofinite topology on .What are the open sets in this topology?What are the closed sets?

A subbasis of a cofinite topology on R is the set of complements of singletons in R. You can get the basis for a cofinite topology on R using the subbasis.What is the basis for the topology?