Connectedness modulo an ideal; a characterization theorem

Research Article

Connectedness modulo an ideal; a characterization theorem

Published in: Quaestiones Mathematicae
Volume 46 , issue 7 , 2023 , pages: 1457–1467
DOI: 10.2989/16073606.2022.2107960
Author(s): M.R. Koushesh Isfahan University of Technology, Iran
Keywords: Primary: 54D05 , Secondary: 03E15 , 46E25 , 54D35 , 54D40 , 54D60 , Stone–Čech compactification , Hewitt realcompactification , connectedness , set ideal , connectedness modulo an ideal
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Abstract

Let X be a topological space and let be an ideal of subsets of X. The space X is called connected modulo if there is no continuous mapping f : X → [0, 1] which is 2-valued modulo in the sense that neither f −1(0) nor f −1(1) belongs to but X \ (f −1(0) ∪ f −1(1)) belongs to . We prove that a completely regular space X is connected modulo if and only if the quotient of the ring CB (X) (of all bounded continuous real-valued mappings on X equipped with pointwise addition and multiplication) is indecomposable. Here is the ideal of CB (X) consisting of all f in CB (X) such that |f|−1([ε, ∞)) belongs to for any positive ε. We examine examples corresponding to various choices of the ideal . We conclude with consideration of the ideal of CB (X) whose importance is highlighted by our characterization theorem.
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