Rings of continuous functions with values in a topological field

Abstract:

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 ${C^\ast }(X,F)$ of all continuous $F$-valued functions with relatively compact range, where the topology of $X$ is similarly determined by ${C^\ast }(X,F)$. 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-Čech 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 Shirota’s theorem concerning the bornologicity of $C(X)$ is obtained.