Research Article
Connectedness modulo an ideal; a characterization theorem
DOI:
10.2989/16073606.2022.2107960
Keywords:
Primary: 54D05, Secondary: 03E15, 46E25, 54D35, 54D40, 54D60, Stone–Čech compactification, Hewitt realcompactification, connectedness, set ideal, connectedness modulo an ideal,
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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