Countable chain condition

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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.

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 separable.

For example,

with the product topology is ccc but not separable.

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