Convergence of Sets? (Topology on a Powerset of a Set?)

The natural topology on $2^U$ is the compact-open topology, which here is the product topology. This is precisely the topology of pointwise convergence of indicator functions $U \to 2$. Thus a sequence $S_1, S_2, ...$ of sets converges in this topology if and only if, for every $u \in U$, either all but finitely many $S_i$ contain $u$ (so that $u$ is in the limit set) or all but finitely many $S_i$ do not contain $u$ (so that $u$ is not in the limit set). So both of the sequences you describe have limit the empty set as desired.

Equivalently (I think), one can define a sequence of sets to converge if its liminf and limsup (defined in the usual way) converge to the same set.