Given a sequence of sets, is there some well-defined notion of a limit of a set?
In other words, given some universe set $U$, I am wondering if there is a topology on $2^U$ (the powerset of $U$) such that the usual intersection and the union limits converge in that topology.
As an explicit example, let $U=\mathbb{N}$, $S_n = \{x\in \mathbb{N} | n< x \le 2n \}$, $T_n = \{n\}$.
The limit of both sequences above should be the empty set by the following argument:
\begin{align} S_n &\subset (n,\infty) \\\\ \lim_{n\to\infty} S_n &\subset \lim_{n\to\infty} (n,\infty) = \cap_{n\in\mathbb{N}} (n,\infty) = \emptyset \end{align}
(I'm not sure how to justify passing a set inclusion to the limit.)