The topology of sets Normally limits come from topologies. However the set-theoretic limit is defined without reference to a specific topology. Therefore I wonder:

What is the topology of sets that corresponds to this limit?

Using this answer to another question of me, I figured out that the set of all subsets with at most $n$ elements of a given set is closed, and that the set of all finite subsets of an infinite set is not closed, but that by itself doesn't really give me an idea what the topology looks like. What I'd like is a general rule when some set is open or closed under this topology.
 A: Identifying sets with their characteristic functions, this is just the product topology.  To be precise, fix a set $X$ and identify subsets of $X$ with their characteristic functions $X\to\{0,1\}$.  Then the topology corresponding to "set-theoretic limits" is just the product topology on the set $\{0,1\}^X$ of all functions $X\to\{0,1\}$, where $\{0,1\}$ has the discrete topology.
Indeed, convergence with respect to the product topology just means pointwise convergence: $(f_n)$ converges to $f$ iff $(f_n(x))$ converges to $f(x)$ for all $x\in X$.  Since $\{0,1\}$ has the discrete topology, this just means that $f_n(x)=f(x)$ for all sufficiently large $n$.  In terms of sets, this means that $(A_n)$ converges to $A$ iff for all $x\in X$, either $x\in A_n$ for all sufficiently large $n$ and $x\in A$, or $x\not\in A_n$ for all sufficiently large $n$ and $x\not\in A$.  This is easily seen to be equivalent to the usual definition of convergence of sets.
(Note that if $X$ is infinite, then there are multiple different topologies on $\{0,1\}^X$ that will give the same notion of convergence of sequences of sets.  However, the product topology is the most natural choice, and is the only one that agrees with the obvious generalization to define convergence of nets of sets.)
