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.

Keywords: closure operator, Galois connection, discrete object, indiscrete object , splitting closure, cosplitting closure, connectedness, disconnectedness

Quaestiones Mathematicae 31(2008), 107–126