The representation theorem

Abstract

If a Boolean algebra A is a field of subsets of a set X,and, in particular, if it is the dual algebra of a Boolean space X, then the points of X serve to define 2-valued homomorphisms on A (see §9). This comment suggests that if we start with a Boolean algebra A and seek to represent it as the dual of some Boolean space X, a reasonable place to conduct the search for points suitable to make up X is among the 2-valued homomorphisms of A. The suggestion would be impractical if it turned out that A has no 2-valued homomorphisms. Our first result along these lines is that there is nothing to fear; there is always a plethora of 2-valued homomorphisms.