Rings of Ordered Field Valued Continuous Functions

The main aim of this paper is to investigate the role of the real field in the theory of Rings of Continuous Functions. In the process the real field has been replaced by an ordered field equipped with its order topology and the function rings of continuous functions on topological spaces are investigated. Apart from generalising the classical theorems of M. H. Stone, Gelfand and Kolmogorov, we have obtained nontrivial generalisations for the classical Banach-Stone Theorem — our version really does not depend on the range of the continuous functions but only depend on the domain topological space. We have also characterised the class of all locally compact non-compact zero dimensional topological spaces as well as the class of all nowhere locally compact zero dimensional spaces using the subrings of functions with compact support. This part of the investigation is carried further to the effect that we have achieved to obtain a kind of Banach-Stone's Theorem for the locally compact non-compact zero dimensional topological spaces using this function ring. The main achievement of this paper is to classify the collection of all zero dimensional but not strongly zero dimensional topological spaces into three kinds, labeled by three inequalities of three compactifications of such spaces