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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.

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    This is not *really* a question fit for [set-theory], I am somewhat reluctant to tag it under [elementary-set-theory] but I cannot think of a better tag.2012-01-22

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I assume the sequence $A_n$ is a sequence of subsets of a set $S$. In that case one can identify the sets $A_n$ with their indicator functions $1_{A_n} : S \to \mathbb{R}$. The space of all functions $S \to \mathbb{R}$ is naturally endowed with the topology of pointwise convergence, and this gives a sensible meaning to the limit of a sequence of indicator functions. I am reasonably certain this limit exists if and only if $\limsup A_n = \liminf A_n$ and it is equal to their common value.

If $S$ has additional structure there are other options. If $S$ is a topological space one can use the compact-open topology on the space of functions $S \to \mathbb{R}$ (this reduces to the above when $S$ is discrete).