Editing 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]]. 

==Overview==
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 (set theory)|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 {{format link|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''', also written as κ-c.c., if every antichain has size less than κ. The countable chain condition is the ℵ<sub>1</sub>-chain condition.

==Examples and properties in topology==
A [[topological space]] is said to satisfy the countable chain condition, or '''[[Mikhail Yakovlevich Suslin|Suslin's]] Condition''', if the partially ordered set of non-empty [[open subset]]s of ''X'' satisfies the countable chain condition, ''i.e.'' every [[pairwise disjoint]] collection of non-empty open subsets of ''X'' is countable. The name originates from  [[Suslin's problem|Suslin's Problem]].

* Every [[separable topological space]] has ccc. Furthermore, a [[Product topology|product space]] of arbitrary amount of separable spaces has ccc.
* A [[metric space]] has ccc if and only if it's separable.
* In general, a topological space with ccc need not be separable. For example, a [[Cantor cube]] <math>\{ 0, 1 \}^\kappa</math> with the [[product topology]] has ccc for any cardinal <math>\kappa</math>, though ''not'' separable for <math>\kappa > \mathfrak{c}</math>.
* Paracompact ccc spaces are [[Lindelöf space|Lindelöf]].
* An example of a topological space with ccc is the real line.

==References==
{{refbegin}}
*{{Citation | last1=Jech | first1=Thomas | author1-link=Thomas Jech | title=Set Theory: Millennium Edition | publisher=[[Springer-Verlag]] | location=Berlin, New York | series=Springer Monographs in Mathematics | isbn=978-3-540-44085-7 | year=2003}}
*Products of Separable Spaces, K. A. Ross, and A. H. Stone. The American Mathematical Monthly 71(4):pp.&nbsp;398–403 (1964)
*Kunen, Kenneth. ''Set Theory: An Introduction to Independence Proofs.''
{{refend}}

[[Category:Order theory]]
[[Category:Forcing (mathematics)]]