# Splitting chains, tunnels and twisted sums

## TL;DR

This paper investigates splitting chains in set theory and their independence from ZFC axioms, demonstrating their applications in constructing special Banach spaces and exploring their topological counterparts called tunnels.

## Contribution

It establishes the independence of splitting chains from ZFC and introduces their use in constructing twisted sums of Banach spaces and in topological tunnel concepts.

## Key findings

- Existence of splitting chains is independent of ZFC.
- Splitting chains can be used to construct twisted sums of Banach spaces.
- Introduction of tunnels as topological analogs of splitting chains.

## Abstract

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.