An infinite subset of a Boolean algebra may fail to have a supremum. (Example: take the finite-cofinite algebra of integers and consider the singletons of the even integers.) A Boolean algebra with the property that every subset of it has both a supremum and an infimum is called a complete (Boolean) algebra. Similarly, a field of sets with the property that both the union and the intersection of every class of sets in the field is again in the field is called a complete field of sets. The simplest example of a complete field of sets (and hence of a complete algebra) is the field of all subsets of a set. Our next example of a complete algebra is not a field; it is the regular open algebra of a topological space (cf. Theorem 1, p. 13). For purposes of reference it is worth while recording the formal statement.
Unable to display preview. Download preview PDF.