  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.
