Splitting chains, tunnels and twisted sums

We study splitting chains in $\mathcal{P}(\omega )$, that is, families of subsets of $\omega$ which are linearly ordered by ${\subseteq }^{\ast }$ and which are splitting. We prove that their existence is independent of axioms of $\mathsf{Z}\mathsf{F}\mathsf{C}$. We show that they can be used to construct certain peculiar Banach spaces: twisted sums of $C({\omega }^{\ast })={\mathrm{\ell }}_{\mathrm{\infty }}\mathrm{/}{c}_0$ and $c_0(\mathfrak c)$. Also, we consider splitting chains in a topological setting, where they give rise to the so called tunnels.