One-dimensional non-Hausdorff manifolds and CW complexes

Abstract

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 $\pi :M\to K\setminus \{v\}$ 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 $f:M\to N$ into a Hausdorff space $N$ there exists a unique continuous map $\hat{f}:K\setminus \{v\}\to N$ such that $f=\hat{f}\circ \pi$.

23 pages, 3 figures. The paper is essentially reworked and we also obtain a characterization of graph-like manifolds