Ordered field valued continuous functions with countable range

• Published in: arXiv.org
• Main Authors: Acharyya, Sudip Kumar, Atasi Deb Ray, Nandi, Pratip
• Format: Paper Journal Article
• Language: English
• Published: Ithaca Cornell University Library, arXiv.org 10.07.2020
• Subjects: Continuity (mathematics); Mathematics - General Topology; Rings (mathematics); Topology
• ISSN: 2331-8422
• DOI: 10.48550/arxiv.2007.05206

For a Hausdorff zero-dimensional topological space \(X\) and a totally ordered field \(F\) with interval topology, let \(C_c(X,F)\) be the ring of all \(F-\)valued continuous functions on \(X\) with countable range. It is proved that if \(F\) is either an uncountable field or countable subfield of \(\mathbb{R}\), then the structure space of \(C_c(X,F)\) is \(\beta_0X\), the Banaschewski Compactification of \(X\). The ideals \(\{O^{p,F}_c:p\in \beta_0X\}\) in \(C_c(X,F)\) are introduced as modified countable analogue of the ideals \(\{O^p:p\in\beta X\}\) in \(C(X)\). It is realized that \(C_c(X,F)\cap C_K(X,F)=\bigcap_{p\in\beta_0X\texttt{\textbackslash}X} O^{p,F}_c\), this may be called a countable analogue of the well-known formula \(C_K(X)=\bigcap_{p\in\beta X\texttt{\textbackslash}X}O^p\) in \(C(X)\). Furthermore, it is shown that the hypothesis \(C_c(X,F)\) is a Von-Neumann regular ring is equivalent to amongst others the condition that \(X\) is a \(P-\)space.