Rings of real-valued continuous functions. I

Abstract

The study, by the last-named author, of the ring of all real-valued continuous functions ~(X, IR) on a tpopological space X, was begun some 33 years ago in the article [1]. The principal notions of this paper are realcompact spaces (originally named Q-spaces for no very good reason) and hyperreal fields (which we call here H-fields). These concepts have found applications in a number of areas. First, topologists have found realcompact spaces to be of interest: see for example [16] and [19]. Second, ultraproducts and ultrapowers, first considered by Log [31], have proved to be a powerful model-theoretic tool for investigation from a unified point of view of varied problems in set theory, logic, model theory, certain branches of algebra, and the theory of numbers (Artin's hypothesis and p-adic fields): see [40] and [8]. The method of ultraproducts and ultrapowers has been worked out in detail, and one may say that a more or less complete theory exists. Its traditional aspects are set forth in the two monographs of Chang and Keisler ([7] and [8]), in the monograph of Comfort and Negrepontis [14], and in the monograph of A.I. Mal'cev [33]. In particular, much progress has been made in such classical problems as the structure of ultrafilters and the cardinal numbers of ultraproducts. The key to the results obtained up to now lies in axiomatic set theory, generic models, and the theory of large cardinal numbers. Studies in recent years have tended to neglect the theory of H-fields. One should not forget, however, that ultrapowers of 1R are exactly the H-fields obtained from discrete spaces. Hence the study of arbitrary H-fields, their algebraic structure, classification, and isomorphism types, should and does lead to more complicated topological and set-theoretical problems. Little progress has been made up to now with