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