Separation versus connectedness

For a closure operator c in the sense of Dikranjan and Giuli, the subcategory Δ(c) (▽(c)) of objects X with c-closed (c-dense) diagonal δX: X → X × X is known to give a general notion of separation (connectedness, respectively), with the expected closure properties under products and subspaces (images), etc. The purpose of this note is to fully characterize the notions of connectedness and disconnectedness in the sense of Arhangelʹskiǐ and Wiegandt and of separation by Pumplün and Röhrl in this context. Briefly, an AW-connectedness is a subcategory of type ▽(c) with c a regular closure operator, and an AW-disconnectedness is of type Δ(c) with c a coregular closure operator, as introduced in this paper. The latter subcategory is in particular PR-separated, i.e., a subcategory of type Δ(c) with c weakly hereditary. Categorical proofs and new applications are provided for the characterization theorems originally given by Arhangelʹskiǐ and Wiegandt in the context of topological spaces.