Connectedness modulo an ideal; a characterization theorem

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 C_B(X) (of all bounded continuous real-valued mappings on X equipped with pointwise addition and multiplication) is indecomposable. Here is the ideal of C_B(X) consisting of all f in C_B(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.

Mathematics Subject Classification (2020):

• Primary: 54D05
• Secondary: 03E15
• 46E25
• 54D35
• 54D40
• 54D60

Key words:

• Stone–Čech compactification
• Hewitt realcompactification
• connectedness
• set ideal
• connectedness modulo an ideal