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