Pairwise disjoint refinements of coverings of one-dimensional Peano continua - Eda, Katsuya; Meilstrup, Mark (Topology Proceedings, 2023)
For a given finite open cover $\mathcal{O}$ of a one-dimensional Peano continuum $X$, there exist refinements $\mathcal{O}_0$ and $\mathcal{O}_1$ of $\mathcal{O}$ consisting of connected open sets such that $\mathcal{O}_0 \cup \mathcal{O}_1$ covers $X$ and both $\mathcal{O}_0$ and $\mathcal{O}_1$ are pairwise disjoint families.