Connectedness modulo an ideal: algebraic characterizations

Abstract

A topological space X is called connected modulo an ideal of its subsets if there exists no continuous function f : X → [0, 1] such that and yet . This paper establishes that X is connected modulo precisely when is Coz-prime (i.e., for cozero-sets A, B ⊆ X satisfying implies , or equivalently, the ring ideal is pseudo-irreducible in C (X ). This result provides a complete characterization of connectedness modulo ideals in terms of topological and algebraic properties. As applications, we unify and simplify several known results while also addressing two unresolved questions in existing research.