Seth Warner

Rings of continuous functions with values in a topological field

Let F be a complete topological field. We undertake a study of the ring C(X, F) of all continuous F-valued functions on a topological space X whose topology is determined by C(X, F), in that it is the weakest making each function in C(X, F) continuous, and of the ring C*(X, F) of all continuous F-valued functions with relatively compact range, where the topology of X is similarly determined by C*(X, F). The theory of uniform structures permits a rapid construction of the appropriate generalizations of the Hewitt realcompactification of X in the former case and of the Stone-Cech compactification of X in the latter. Most attention is given to the case where F and X are ultraregular; in this case we determine conditions on F that permit a development parallel to the classical theory where F is the real number field. One example of such conditions is that the cardinality of F be nonmeasurable and that the topology of F be given by an ultrametric or a valuation. Measure-theoretic interpretations are given, and a nonarchimedean analogue of Nachbin and Shirotas theorem concerning the bornologicity of C(X) is obtained. Our goal is to develop a theory parallel to that of realcompact spaces and realcompactifications by replacing the topological field R of real numbers with a complete topological field F, with emphasis on the case where F is ultraregular. (An ultraregular space is a Hausdorff space for which the clopen sets form a basis, a clopen set being one that is both open and closed. An ultranormal space is a Hausdorff space each closed subset of which has a fundamental system of neighborhoods consisting of clopen sets. There exist ultraregular spaces that are not ultranormal [25], [261.) Our emphasis on ultraregular fields is prompted by the fact that every field whose topology is given by a valuation or an ultrametric is ultraregular (and, in fact, is ultranormal). If X and Y are topological spaces, C(X, Y) denotes the set of all continuous functions from X into Y, and C*(X, Y) denotes the set of all continuous functions f from X into Y such that f(X) is compact. If F is a topological field, C(X, F) and C*(X, F) are both F-algebras; if, moreover, F is locally compact, Received by the editors July 24, 1973. AMS (MOS) subject classifications (1970). Primary 54D35, 54D60, 54E15.