Suppose we have a finite $\sigma$ -field S, and that sets A and B $\in$ S. Since S is closed under union & complementation [by def] $(A' \cup B')' = (A \cap B)'$ and $(A \cap B)$ lie in S as well. This means that the intersection of any 2 sets in S lie in S as well.

Doesn't this make S a topology as well? So does it follow that every finite $\sigma$ -field is a topology?