Splitting chains, tunnels and twisted sums

Abstract

We study splitting chains in $$\mathcal{P}(\omega)$$ , that is, families of subsets of ω which are linearly ordered by ⊆* and which are splitting. We prove that their existence is independent of axioms of ZFC. We show that they can be used to construct certain peculiar Banach spaces: twisted sums of C(ω*) = l∞/c0 and c0(c). Also, we consider splitting chains in a topological setting, where they give rise to the so called tunnels.