Constructing Banaschewski compactification without Dedekind completeness axiom

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

A generalized version of the rings $C_K(X)$ and $C_\infty(X)$ - an enquery about when they become Noetherian

“…Their purpose is to look into a few aspects on the possible interplay between the topological structure on X and the algebraic structure of C(X, F ) and their subrings mentioned in the last sentence. It was observed in the same paper [2] that a CFR-space with F not isomorphic to R is zero-dimensional, in particular Tychonoff. Conversely each zero-dimensional Hausdorff topological space becomes CFR-space for any ordered field F .…”

“…We get the familiar ring C(X) on choosing F = R. X is called completely F -regular (CFR in short) if given a closed set K in X and a point x ∈ X −K, there exists an f ∈ C(X, F ) such that f (x) = 0 and f (K) = 1. The ring C(X, F ) together with a few of its subrings were investigated by Acharyya, Chattopadhyaya and Ghosh [2]. Their purpose is to look into a few aspects on the possible interplay between the topological structure on X and the algebraic structure of C(X, F ) and their subrings mentioned in the last sentence.…”

A class of ideals in intermediate rings of continuous functions

“…[Theorem 2.7 [2]] For any topological space X and for any totally ordered field F , the zero sets in X with respect to the field F are G δ subsets of X if and only if cf(F)= ω • . Lemma 3.13.…”

Continuous functions with compact support

“…The rings of continuous functions on a topological space X with values in a linearly ordered field, equipped with its order topology, was initiated in the paper [4]. The purpose of this paper is to investigate a lattice ordered commutative subring of C * (X, F ), see Definition 2.1, which consists of precisely those continuous functions defined on a topological space X and taking values in a linearly ordered field equipped with its order topology that have a compact support.…”

“…(1) we can provide characterizations for completely F regular locally compact non-compact topological spaces (see Theorem 3.4) as well as completely F regular nowhere locally compact topological spaces (see Theorem 3.5), (2) develop an analogue of the idea of structure spaces so as to determine the class of locally compact non-compact Tychonoff topological spaces in terms of these function rings (see Theorem 4.9), thereby generalizing the Banach Stone Theorem, see [3,Theorem 4.9, page 57]. The paper is organized as follows : §2 introduces the basic notions and results from [4] that are required to make the paper self-contained. §3 develops the basic properties of the function ring C k (X, F ) culminating in the characterizations stated above.…”