Complete Boolean Algebra - Examples

Examples

  • 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.
  • The algebra of all measurable subsets of a measure space is a ℵ1-complete Boolean algebra, but is not usually complete.
  • 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 Boolean algebra of all Baire sets modulo meager sets in a topological space with a countable base is 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. The resulting object, denoted P(ω)/Fin, consists of all equivalence classes of sets of naturals, where the relevant equivalence relation is that two sets of naturals are equivalent if their symmetric difference is finite. The Boolean operations are defined analogously, for example, if A and B are two equivalence classes in P(ω)/Fin, we define to be the equivalence class of, where a and b are some (any) elements of A and B respectively.
Now let a0, a1,... be pairwise disjoint infinite sets of naturals, and let A0, A1,... be their corresponding equivalence classes in P(ω)/Fin . Then given any upper bound X of A0, A1,... in P(ω)/Fin, we can find a lesser upper bound, by removing from a representative for X one element of each an. Therefore the An have no supremum.
  • A Boolean algebra is complete if and only if its Stone space of prime ideals is extremally disconnected.

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