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- Countable_chain_condition abstract "In order theory, a partially ordered set X is said to satisfy the countable chain condition, or to be ccc, if every strong antichain in X is countable. There are really two conditions: the upwards and downwards countable chain conditions. These are not equivalent. The countable chain condition means the downwards countable chain condition, in other words no two elements have a common lower bound.This is called the \"countable chain condition\" rather than the more logical term \"countable antichain condition\" for historical reasons related to certain chains of open sets in topological spaces and chains in complete Boolean algebras, where chain conditions sometimes happen to be equivalent to antichain conditions. For example, if κ is a cardinal, then in a complete Boolean algebra every antichain has size less than κ if and only if there is no descending κ-sequence of elements, so chain conditions are equivalent to antichain conditions.Partial orders and spaces satisfying the ccc are used in the statement of Martin's axiom.In the theory of forcing, ccc partial orders are used because forcing with any generic set over such an order preserves cardinals and cofinalities. Furthermore, the c.c.c. property is preserved by finite support iterations (see iterated forcing). For more information on ccc in the context of forcing, see Forcing (set theory) § The countable chain condition.More generally, if κ is a cardinal then a poset is said to satisfy the κ-chain condition if every antichain has size less than κ. The countable chain condition is the ℵ1-chain condition.".
- Countable_chain_condition wikiPageID "1550771".
- Countable_chain_condition wikiPageLength "3053".
- Countable_chain_condition wikiPageOutDegree "19".
- Countable_chain_condition wikiPageRevisionID "698112659".
- Countable_chain_condition wikiPageWikiLink Cardinality_of_the_continuum.
- Countable_chain_condition wikiPageWikiLink Category:Forcing_(mathematics).
- Countable_chain_condition wikiPageWikiLink Category:Order_theory.
- Countable_chain_condition wikiPageWikiLink Countable_set.
- Countable_chain_condition wikiPageWikiLink Disjoint_sets.
- Countable_chain_condition wikiPageWikiLink Forcing_(mathematics).
- Countable_chain_condition wikiPageWikiLink Iterated_forcing.
- Countable_chain_condition wikiPageWikiLink Martins_axiom.
- Countable_chain_condition wikiPageWikiLink Metric_space.
- Countable_chain_condition wikiPageWikiLink Open_set.
- Countable_chain_condition wikiPageWikiLink Order_theory.
- Countable_chain_condition wikiPageWikiLink Partially_ordered_set.
- Countable_chain_condition wikiPageWikiLink Product_topology.
- Countable_chain_condition wikiPageWikiLink Separable_space.
- Countable_chain_condition wikiPageWikiLink Springer_Science+Business_Media.
- Countable_chain_condition wikiPageWikiLink Strong_antichain.
- Countable_chain_condition wikiPageWikiLink Suslins_problem.
- Countable_chain_condition wikiPageWikiLink Topological_space.
- Countable_chain_condition wikiPageWikiLinkText "Countable chain condition".
- Countable_chain_condition wikiPageWikiLinkText "ccc".
- Countable_chain_condition wikiPageWikiLinkText "countable chain condition".
- Countable_chain_condition wikiPageUsesTemplate Template:Citation.
- Countable_chain_condition wikiPageUsesTemplate Template:Format_link.
- Countable_chain_condition subject Category:Forcing_(mathematics).
- Countable_chain_condition subject Category:Order_theory.
- Countable_chain_condition type Field.
- Countable_chain_condition comment "In order theory, a partially ordered set X is said to satisfy the countable chain condition, or to be ccc, if every strong antichain in X is countable. There are really two conditions: the upwards and downwards countable chain conditions. These are not equivalent.".
- Countable_chain_condition label "Countable chain condition".
- Countable_chain_condition sameAs Q5176824.
- Countable_chain_condition sameAs Condition_de_chaîne_dénombrable.
- Countable_chain_condition sameAs 可算鎖条件.
- Countable_chain_condition sameAs m.059p1d.
- Countable_chain_condition sameAs Q5176824.
- Countable_chain_condition wasDerivedFrom Countable_chain_condition?oldid=698112659.
- Countable_chain_condition isPrimaryTopicOf Countable_chain_condition.