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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. We adopt the convention 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 complete Boolean algebras. (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. 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 wikiPageRevisionID "586465212".
- Countable_chain_condition hasPhotoCollection Countable_chain_condition.
- Countable_chain_condition subject Category:Forcing_(mathematics).
- Countable_chain_condition subject Category:Order_theory.
- 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. We adopt the convention the countable chain condition means the downwards countable chain condition, in other words no two elements have a common lower bound.".
- Countable_chain_condition label "Condition de chaîne dénombrable".
- Countable_chain_condition label "Countable chain condition".
- Countable_chain_condition label "可算鎖条件".
- 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 sameAs Q5176824.
- Countable_chain_condition wasDerivedFrom Countable_chain_condition?oldid=586465212.
- Countable_chain_condition isPrimaryTopicOf Countable_chain_condition.