期刊名称：International Journal of Mathematics and Mathematical Sciences
印刷版ISSN：0161-1712
电子版ISSN：1687-0425
出版年度：2004
卷号：2004
DOI：10.1155/S0161171204305168
出版社：Hindawi Publishing Corporation
摘要：The main aim of this paper is to provide a construction of the Banaschewski compactification of a zero-dimensional Hausdorff topological space as a structure space of a ring of ordered
field-valued continuous functions on the space, and thereby exhibit the independence of the construction from any
completeness axiom for an ordered field. In the process of describing this construction we have generalized the classical versions of M. H. Stone's theorem, the Banach-Stone theorem, and
the Gelfand-Kolmogoroff theorem. The paper is concluded with a conjecture of a split in the class of all
zero-dimensional but not strongly zero-dimensional Hausdorff topological spaces into three classes that are labeled by
inequalities between three compactifications of X, namely, the
Stone-Čech compactification βX, the Banaschewski
compactification β0X, and the structure space
𝔐X,F of the lattice-ordered commutative ring ℭ(X,F) of all continuous functions on X taking values in the ordered
field F, equipped with its order topology. Some open problems
are also stated.
