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- Directed_set abstract "In mathematics, a directed set (or a directed preorder or a filtered set) is a nonempty set A together with a reflexive and transitive binary relation ≤ (that is, a preorder), with the additional property that every pair of elements has an upper bound. In other words, for any a and b in A there must exist a c in A with a ≤ c and b ≤ c.The notion defined above is sometimes called an upward directed set. A downward directed set is defined analogously, meaning when every doubleton is bounded below. Some authors (and this article) assume that a directed set is directed upward, unless otherwise stated. Beware that other authors call a set directed if and only if it is directed both upward and downward.Directed sets are a generalization of nonempty totally ordered sets, that is, all totally ordered sets are directed sets (contrast partially ordered sets which need not be directed). Join semilattices (which are partially ordered sets) are directed sets as well, but not conversely. Likewise, lattices are directed sets both upward and downward.In topology, directed sets are used to define nets, which generalize sequences and unite the various notions of limit used in analysis. Directed sets also give rise to direct limits in abstract algebra and (more generally) category theory.".
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- Directed_set wikiPageLength "7790".
- Directed_set wikiPageOutDegree "46".
- Directed_set wikiPageRevisionID "683330595".
- Directed_set wikiPageWikiLink Abstract_algebra.
- Directed_set wikiPageWikiLink Antisymmetric_relation.
- Directed_set wikiPageWikiLink Binary_relation.
- Directed_set wikiPageWikiLink Category:General_topology.
- Directed_set wikiPageWikiLink Category:Mathematical_relations.
- Directed_set wikiPageWikiLink Category:Order_theory.
- Directed_set wikiPageWikiLink Category_theory.
- Directed_set wikiPageWikiLink Centered_set.
- Directed_set wikiPageWikiLink Complete_partial_order.
- Directed_set wikiPageWikiLink Coordinatewise_order.
- Directed_set wikiPageWikiLink Direct_limit.
- Directed_set wikiPageWikiLink Directed_complete_partial_order.
- Directed_set wikiPageWikiLink Domain_theory.
- Directed_set wikiPageWikiLink Empty_set.
- Directed_set wikiPageWikiLink Filter_(mathematics).
- Directed_set wikiPageWikiLink Filtered_category.
- Directed_set wikiPageWikiLink Ideal_(order_theory).
- Directed_set wikiPageWikiLink Infimum_and_supremum.
- Directed_set wikiPageWikiLink Lattice_(order).
- Directed_set wikiPageWikiLink Least_upper_bound.
- Directed_set wikiPageWikiLink Limit_(mathematics).
- Directed_set wikiPageWikiLink Linked_set.
- Directed_set wikiPageWikiLink Lower_set.
- Directed_set wikiPageWikiLink Mathematical_analysis.
- Directed_set wikiPageWikiLink Mathematics.
- Directed_set wikiPageWikiLink Natural_number.
- Directed_set wikiPageWikiLink Neighbourhood_(mathematics).
- Directed_set wikiPageWikiLink Net_(mathematics).
- Directed_set wikiPageWikiLink Net_(topology).
- Directed_set wikiPageWikiLink Partial_order.
- Directed_set wikiPageWikiLink Partially_ordered_set.
- Directed_set wikiPageWikiLink Partially_ordered_sets.
- Directed_set wikiPageWikiLink Poset.
- Directed_set wikiPageWikiLink Preorder.
- Directed_set wikiPageWikiLink Product_order.
- Directed_set wikiPageWikiLink Real_number.
- Directed_set wikiPageWikiLink Reflexive_relation.
- Directed_set wikiPageWikiLink Semilattice.
- Directed_set wikiPageWikiLink Sequence.
- Directed_set wikiPageWikiLink Set_(mathematics).
- Directed_set wikiPageWikiLink Topological_neighbourhood.
- Directed_set wikiPageWikiLink Topological_space.
- Directed_set wikiPageWikiLink Topology.
- Directed_set wikiPageWikiLink Total_order.
- Directed_set wikiPageWikiLink Totally_ordered_set.
- Directed_set wikiPageWikiLink Transitive_relation.
- Directed_set wikiPageWikiLink Upper_and_lower_bounds.
- Directed_set wikiPageWikiLink Upper_bound.
- Directed_set wikiPageWikiLink Upper_set.
- Directed_set wikiPageWikiLink File:Directed_set,_but_no_join_semi-lattice.png.
- Directed_set wikiPageWikiLinkText "Directed Set".
- Directed_set wikiPageWikiLinkText "Directed set".
- Directed_set wikiPageWikiLinkText "Directed".
- Directed_set wikiPageWikiLinkText "Directed_set".
- Directed_set wikiPageWikiLinkText "directed set".
- Directed_set wikiPageWikiLinkText "directed subset".
- Directed_set wikiPageWikiLinkText "directed subsets".
- Directed_set wikiPageWikiLinkText "directed".
- Directed_set wikiPageWikiLinkText "upward-directed".
- Directed_set hasPhotoCollection Directed_set.
- Directed_set wikiPageUsesTemplate Template:Reflist.
- Directed_set subject Category:General_topology.
- Directed_set subject Category:Mathematical_relations.
- Directed_set subject Category:Order_theory.
- Directed_set hypernym Nonempty.
- Directed_set type Concept.
- Directed_set type Field.
- Directed_set type Relation.
- Directed_set comment "In mathematics, a directed set (or a directed preorder or a filtered set) is a nonempty set A together with a reflexive and transitive binary relation ≤ (that is, a preorder), with the additional property that every pair of elements has an upper bound. In other words, for any a and b in A there must exist a c in A with a ≤ c and b ≤ c.The notion defined above is sometimes called an upward directed set.".
- Directed_set label "Directed set".
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- Directed_set sameAs Gerichtete_Menge.
- Directed_set sameAs Ensemble_ordonné_filtrant.
- Directed_set sameAs Insieme_diretto.
- Directed_set sameAs 有向集合.
- Directed_set sameAs 유향_집합.
- Directed_set sameAs Zbiór_skierowany.
- Directed_set sameAs Conjunto_direcionado.
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- Directed_set sameAs Направленное_множество.
- Directed_set sameAs Направлена_множина.
- Directed_set sameAs Tập_được_định_hướng.
- Directed_set sameAs Q1513048.
- Directed_set sameAs Q1513048.
- Directed_set sameAs 有向集合.
- Directed_set wasDerivedFrom Directed_set?oldid=683330595.
- Directed_set isPrimaryTopicOf Directed_set.