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International Journal of Mathematics and Mathematical Sciences
Volume 2004, Issue 69, Pages 3799-3816
http://dx.doi.org/10.1155/S0161171204305168

Constructing Banaschewski compactification without Dedekind completeness axiom

1Department of Pure Mathematics, University of Calcutta, 35 Ballygaunge Circular Road, Calcutta, West Bengal 700 019, India
2Department of Mathematics, University of Burdwan, Burdwan 713 104, West Bengal, India
3School of Mathematical and Statistical Sciences, Howard College Campus, University of KwaZulu-Natal, Durban 4041, South Africa
4Department of Mathematics and Applied Mathematics, University of Cape Town, Rondebosch 7701, Cape Town, South Africa

Received 14 May 2003

Copyright © 2004 Hindawi Publishing Corporation. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

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 (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.