a set that joins these accumulation sets together. That is, it is the union of all of the accumulation sets. When ordering by set inclusion, the supremum limit is the least upper bound on the set of accumulation points because it contains each of them. Hence, it is the supremum of the limit points. The infimum/inferior/inner limit is a set where all of these accumulation sets meet. That is, it is the intersection of all of the accumulation sets. When ordering by set inclusion, the infimum limit is the greatest lower bound on the set of accumulation points because it is contained in each of them. Hence, it is the infimum of the limit points. Because ordering is by set inclusion, then the outer limit will always contain the inner limit (i.e., lim inf Xn ⊆ lim sup Xn). Hence, when considering the convergence of a sequence of sets, it generally suffices to consider the convergence of the outer limit of that sequence. The difference between the two definitions involves how the topology (i.e., how to quantify separation) is defined. In fact, the second definition is identical to the first when the discrete metric is used to induce the topology on X. General set convergence A sequence of sets in a metrizable space approaches a limiting set when the elements of each member of the sequence approach the elements of the limiting set. In particular, if is a sequence of subsets of then: which is also called the outer limit, consists of those elements which are limits of points in taken from (countably) infinitely many That is, if and only if there exists a sequence of points and a of such that and which is also called the inner limit, consists of those elements which are limits of points in for all but finitely many (that is, cofinitely many ). That is, if and only if there exists a of points such that and The limit exists if and only if and agree, in which case The outer and inner limits should not be confused with the set-theoretic limits superior and inferior, as the latter sets are not sensitive to the topological structure of the space. Special case: discrete metric This is the definition used in measure theory and probability. Further discussion and examples from the set-theoretic point of view, as opposed to the topological point of view discussed below,