Complete Boolean algebra

In mathematics, a complete Boolean algebra is a Boolean algebra in which every subset has a supremum (least upper bound). Complete Boolean algebras are used to construct Boolean-valued models of set theory in the theory of forcing. Every Boolean algebra A has an essentially unique completion, which is a complete Boolean algebra containing A such that every element is the supremum of some subset of A. As a partially ordered set, this completion of A is the Dedekind–MacNeille completion. More generally, if κ is a cardinal then a Boolean algebra is called κ-complete if every subset of cardinality less than κ has a supremum. Every finite Boolean algebra is complete. The algebra of subsets of a given set is a complete Boolean algebra. The regular open sets of any topological space form a complete Boolean algebra. This example is of particular importance because every forcing poset can be considered as a topological space (a base for the topology consisting of sets that are the set of all elements less than or equal to a given element). The corresponding regular open algebra can be used to form Boolean-valued models which are then equivalent to generic extensions by the given forcing poset. The algebra of all measurable subsets of a σ-finite measure space, modulo null sets, is a complete Boolean algebra. When the measure space is the unit interval with the σ-algebra of Lebesgue measurable sets, the Boolean algebra is called the random algebra. The Boolean algebra of all Baire sets modulo meager sets in a topological space with a countable base is complete; when the topological space is the real numbers the algebra is sometimes called the Cantor algebra. The algebra of all subsets of an infinite set that are finite or have finite complement is a Boolean algebra but is not complete. The algebra of all measurable subsets of a measure space is a א1-complete Boolean algebra, but is not usually complete. Another example of a Boolean algebra that is not complete is the Boolean algebra P(ω) of all sets of natural numbers, quotiented out by the ideal Fin of finite subsets.