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chain condition (Definition)

A partial order $ P$ satisfies the $ \kappa$-chain condition if for any $ S\subseteq P$ with $ \vert S\vert=\kappa$ then there exist distinct $ x,y\in S$ and some $ p$ such that $ p\leq x$ and $ p\leq y$.

If $ \kappa=\aleph_1$ then $ P$ is said to satisfy the countable chain condition (c.c.c.)



"chain condition" is owned by Henry.
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See Also: partial order, partial order with chain condition does not collapse cardinals

Also defines:  chain condition, countable chain condition, $\kappa$-chain condition, c.c.c., ccc
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Cross-references: satisfies, partial order
There are 10 references to this entry.

This is version 4 of chain condition, born on 2002-07-30, modified 2006-06-23.
Object id is 3241, canonical name is ChainCondition.
Accessed 7080 times total.

Classification:
AMS MSC: 03E35 (Mathematical logic and foundations :: Set theory :: Consistency and independence results)
Pending Errata and Addenda
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omega_1 ? by AxelBoldt on 2002-08-12 11:52:18
Maybe omega_1 should be aleph_1 in the definition of c.c.c, since we really need a cardinal number here.
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