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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Rings of continuous functions with values in a topological field

Authors: George Bachman, Edward Beckenstein, Lawrence Narici and Seth Warner
Journal: Trans. Amer. Math. Soc. 204 (1975), 91-112
MSC: Primary 54D35; Secondary 54C35
MathSciNet review: 0402687
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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.

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Keywords: Topological field, realcompactification, ultraregular, ultranormal
Article copyright: © Copyright 1975 American Mathematical Society