Connectedness modulo an ideal; a characterization theorem

Abstract

Let X be a topological space and let I be an ideal of subsets of X. The space X is called connected modulo I if there is no continuous mapping f : X à [0; 1] which is 2-valued modulo I in the sense that neither f^-1(0) nor f^-1 (1) belongs to I but X \ (f^-1(0) [ f^-1 (1)) belongs to I. We prove that a completely regular space X is connected modulo I if and only if the quotient C_B(X)=C^I₀ (X) of the ring C_B(X) (of all bounded continuous real-valued mappings on X equipped with pointwise addition and multiplication) is indecomposable. Here C C^I₀ (X) is the ideal of C_B(X) consisting of all f in C_B(X) such that │f│^-1([ϵ; ∞)) belongs to I for any positive ϵ. We examine examples corresponding to various choices of the ideal I. We conclude with consideration of the ideal C^I₀ (X) of C_B(X) whose importance is highlighted by our characterization theorem.