1. Directed sets are a generalization of nonempty totally ordered sets.2. In topology, directed sets are used to define analysis. 3. Observe that the scale is by definition a directed set . 4. Any directed set " A " may be made into a Cauchy space. 5. The notion defined above is sometimes called an "'upward directed set " '. 6. Nets generalize the notion of a sequence by requiring the index set simply be a directed set . 7. Directed sets also give rise to direct limits in abstract algebra and ( more generally ) category theory.8. These are posets in which every upward-directed set is required to have a least upper bound. 9. On the other hand, consider the directed set ( in fact : the chain ) of finite sets 10. Some authors ( and this article ) assume that a directed set is directed upward, unless otherwise stated.
