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 ccc property is preserved by finite support iterations (see
iterated forcing). For more information on ccc in the context of forcing, see . More generally, if
κ is a cardinal then a poset is said to satisfy the '''
κ-chain condition'
, also written as κ
-c.c., if every strong antichain has size less than κ''. The countable chain condition is the ℵ1-chain condition. ==Examples and properties in topology==