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Countable chain condition || in good condition

Published September 18th, 2007 in Uncategorized

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,

<math>\{ 0, 1 \}^{ 2^{ 2^{ \aleph_0 }\ } }</math>

with the product topology is ccc but not separable.

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