Splitting chains, tunnels and twisted sums

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