I would like to better understand limits of sets.
Suppose we have a sequence $(A_n)$ of sets and we would like to study its behaviour as $n \to \infty$.
Do we need to assume the sequence of sets is monotone in order for the question of limit behaviour to make sense ?
In case the sequence is increasing (decreasing), do we define the limit as the union (intersection)? Or do we define the limit (in case it exists) to be equal to $\limsup A_n = \liminf A_n$ ?
many thanks.