Applications of the theory of Boolean rings to general topology

In an earlier paperf we have developed an abstract theory of Boolean algebras and their representations by algebras of classes. We now relate this theory to the study of general topology. The first part of our discussion is devoted to showing that the theory of Boolean rings is mathematically equivalent to the theory of locally-bicompact totally-disconnected topological spaces. In R we have already prepared the way for a topological treatment of the perfect representation of an arbitrary Boolean ring. Continuing in this way, we find that the perfect representation is converted by the introduction of a suitable topology into a space of the indicated type. We have no difficulty in inverting this result, proving that every locally-bicompact totally-disconnected topological space arises by the same procedure from a suitable Boolean ring.' It is thus convenient to call the spaces corresponding in this manner to Boolean rings, Boolean spaces. The algebraic properties of Boolean rings can, of course, be correlated in detail with the topological properties of the corresponding Boolean spaces. A simple instance of the correlation is the theorem that the Boolean rings with unit are characterized as those for which the corresponding Boolean spaces are bicompact. A familiar example of a bicompact Boolean space is the Cantor discontinuum or ternary set, which we discuss at the close of Chapter I. Having established this direct connection between Boolean rings and topology, we proceed in the second part of the discussion to considerations of a yet more general nature. We propose the problem of representing an arbitrary TVspace by means of maps in bicompact Boolean spaces. Our solution of this problem embodies an explicit construction of such maps, which we shall now describe briefly. In a given TVspace dt, the open sets and the nowhere dense sets generate a Boolean ring, with 9Î as unit, which characterizes the topological structure of 9Î. Those subrings which contain 9Î and which are so large that the interiors of their member sets constitute bases for 9î, also char-