are at set-theoretic limit. By this definition, a sequence of sets approaches a limiting set when the limiting set includes elements which are in all except finitely many sets of the sequence and does not include elements which are in all except finitely many complements of sets of the sequence. That is, this case specializes the general definition when the topology on set X is induced from the discrete metric. Specifically, for points x, y ∈ X, the discrete metric is defined by under which a sequence of points (xk) converges to point x ∈ X if and only if xk = x for all but finitely many k. Therefore, if the limit set exists it contains the points and only the points which are in all except finitely many of the sets of the sequence. Since convergence in the discrete metric is the strictest form of convergence (i.e., requires the most), this definition of a limit set is the strictest possible. If (Xn) is a sequence of subsets of X, then the following always exist: lim sup Xn consists of elements of X which belong to Xn for infinitely many n (see countably infinite). That is, x ∈ lim sup Xn if and only if there exists a subsequence (Xnk) of (Xn) such that x ∈ Xnk for all k. lim inf Xn consists of elements of X which belong to Xn for all except finitely many n (i.e., for cofinitely many n). That is, x ∈ lim inf Xn if and only if there exists some m > 0 such that x ∈ Xn for all n > m. Observe that x ∈ lim sup Xn if and only if x ∉ lim inf Xnc. lim Xn exists if and only if lim inf Xn and lim sup Xn agree, in which case lim Xn = lim sup Xn = lim inf Xn. In this sense, the sequence has a limit so long as every point in X either appears in all except finitely many Xn or appears in all except finitely many Xnc. Using the standard parlance of set theory, set inclusion provides a partial ordering on the collection of all subsets of X that allows set intersection to generate a greatest lower bound and set union to generate a least upper bound. Thus, the infimum or meet of a collection of subsets is the greatest lower