Countable chain condition

See also: Forcing (set theory)#The countable chain condition

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.)

A topological space is said to satisfy the countable chain condition if the partially ordered set of non-empty open subsets of X satisfies the countable chain condition, i.e. if every pairwise disjoint collection of non-empty open subsets of X is countable.

Every separable topological space is ccc. Every metric space which is ccc is also separable, but in general a ccc topological space need not be separable.

For example,

$\{ 0, 1 \}^{ 2^{ 2^{ \aleph_0 }\ } }$

with the product topology is ccc but not separable.

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.

References

• Jech, Thomas (2003), Set Theory: Millennium Edition, Springer Monographs in Mathematics, Berlin, New York: Springer-Verlag, ISBN 978-3-540-44085-7