Applications of the theory of Boolean rings to general topology

Abstract

The notations of the present paper are largely determined by the usages already established in R. Thus, for instance, in considering Boolean rings as abstract entities, we continue to use the symbols -, v , +, <, ' corresponding to the symbols ■, u , A, c, ', of the theory of classes.In the algebra of the closure operation, we write A~ in place of A for convenience in putting down the more complicated expressions such as A~'~'.The contents of the present paper may be summarized systematically under the following headings: Chapter I, Boolean spaces: §1, Introduction of topological concepts; §2, Relations between algebra and topology; §3, Universal Boolean rings and spaces.Chapter II, Maps in Boolean spaces: §1, The general theory of maps; §2, Construction of Boolean maps; §3, Relation between algebraic and other maps; §4, Applications to the theory of extensions; §5, Totally-disconnected and discrete spaces.Chapter III, «Stronger separation conditions: §1