One-dimensional non-Hausdorff manifolds and CW complexes
Say that a connected non-Hausdorff one-dimensional manifold $M$ is "graph-like", whenever the set $M_{br}$ of its non-Hausdorff (called "branch") points is locally finite and every connected component of its complement has a countable base. We prove that for every graph-like manifold $M$ there exists a one-dimensional CW complex $K$ with the set of vertices $K^{(0)}$, a vertex $v\in K^{(0)}$ and a (surjective) quotient map $\pi\colon M \to K\setminus\{v\}$ such that $\pi^{-1}(K^{(0)}\setminus \{v\}) = M_{br}\cup\partial M$. Conversely, existence of such quotient map $\pi$ and the assumption that $M_{br}$ is nowhere dense imply that $M$ is graph-like. Moreover, in this case $K\setminus\{v\}$ is the minimal Hausdorff factor of $M$, that is, for every continuous map $f\colon M \to N$ into a Hausdorff space $N$ there exists a unique continuous map $\hat{f}\colon K\setminus\{v\}\to N$ such that $f = \hat{f}\circ\pi$.