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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.)

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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.

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    @Zhen: in a category with finite coproducts and a terminal object,$2$is sometimes used to denote the coproduct of the terminal object with itself. In Top, this is the 2-point space with the discrete topology. (Alternately, the forgetful functor from Top to Set has two adjoints: its left adjoint gives a set the discrete topology and its right adjoint gives a set the indiscrete topology. Which adjoint is appropriate for your purposes depends, of course, on the purposes.)2011-01-24