Separation versus Connectedness
Abstract: For a closure operator c in the sense of Dikranjan and Giuli, the subcategory
Delta(c) (Nabla(c)) of objects X with c-closed (c-dense)
diagonal delta_X:X --> 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
Arhangel'skii and Wiegandt and of separation by Pumplun and
Rohrl in this context.
Briefly, an AW-connectedness is a subcategory of type Nabla(c) with c
a regular closure operator, and an AW-disconnectedness is of type
Delta(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 Delta(c) with c weakly hereditary.
Categorical proofs and new applications are provided for the characterization
theorems originally given by Arhangel'skii and Wiegandt in the
context of topological spaces.
