International Journal of Mathematics and Mathematical Sciences
Volume 2004 (2004), Issue 69, Pages 3799-3816
doi:10.1155/S0161171204305168
Constructing Banaschewski compactification without Dedekind completeness axiom
1Department of Pure Mathematics, University of Calcutta, 35 Ballygaunge Circular Road, Calcutta 700 019, West Bengal, 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 S. K. Acharyya et al. 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 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.