Connectednesses and disconnectednesses in topology

AbstractThe development of a general theory for connectednesses and disconnectednesses of topological spaces was started in Preuss' Ph.D. thesis [8], in [4], [9], [10] and [11]. Preuss' investigations exhibited the until then hidden, but conjectured relationship between separation axioms and not-connectedness of topological spaces. Our present paper clarifies the fact that the theory of connectednesses and disconnectednesses of topological spaces corresponds to the radical—semisimple theory of rings and to the torsion—torsionfree theory of abelian categories (cf. Theorems 2.1, and 3.1).It is the purpose of this paper to study and characterize disconnectednesses and connectednesses of topological spaces. Disconnectednesses will be characterized as hereditary, productive and upwards-closed classes and connectednesses will be classified as continuously closed, second additive and q-reversible classes of topological spaces. It turns out that the class of totally disconnected spaces is the smallest non-trivial disconnectedness; the largest and second largest non-trivial disconnectedness is the class of T0-and T1-spaces, respectively. It is shown that there are infinitely many disconnectednesses and they do not form a chain. Finally some questions concerning the theory of connectednesses and disconnectednesses are stated