A coherent family of partial functions on $\mathbb {N}$

We prove that there is a family of partial functions $f_\alpha :A_\alpha \to \alpha$ $(\alpha \to \omega _1,A_\alpha$ is a tower in $P(\omega )/\operatorname {Fin})$ such that every surjection $g:\omega _1\to \{0,1\}$ is associated to a cohomologically different Hausdorff gap (see Talayco). This improves a result of Talayco.