11. As in the case of limits of sequences, least upper bounds of directed sets do not always exist. 12. That is, all totally ordered sets are directed sets ( contrast lattices are directed sets both upward and downward. 13. That is, all totally ordered sets are directed sets ( contrast lattices are directed sets both upward and downward. 14. Also note that, by considering directed sets of two elements, such a function also has to be monotonic. 15. Cofinal subsets are very important in the theory of directed sets and cofinal subnet is the appropriate generalization of subsequence . 16. A "'downward directed set "'is defined analogously, meaning when every pair of elements is bounded below. 17. A "'net "'is a function from a ( possibly uncountable ) directed set to a topological space. 18. Now, as in the case of sequences, we are interested in the " limit " of a directed set . 19. Every sequence is a net, taking " A " to be the directed set of natural numbers with the usual ordering. 20. In the formalization of order theory, this is just the "'least upper bound "'of the directed set .
