Discrete Objects, Splitting Closure and Connectedness

Abstract

Notions of discrete and indiscrete classes with respect to a closure operator are introduced and studied. These notions are strongly related to splitting and cosplitting closure operators. By linking the above concepts, two Galois connections arise whose composition provides a third Galois connection that can be used as a common generalization of connectedness and disconnectedness in topology and torsion theories in algebra. Examples are provided.