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. Note that there are really two conditions here: The upwards and downwards countable chain conditions. These are not equivalent. We are adopting the convention the countable chain condition means the downwards countable chain condition.

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.

Note that 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 seperable.

For example, <math>\{ 0, 1 \}^{ 2^{ 2^{\aleph_0} } }<math> with the product topology is ccc but not seperable.

ccc partial orders and spaces are of most interest when discussing Martin's Axiom.