One-dimensional non-Hausdorff manifolds and CW complexes

Abstract

This paper studies one-dimensional non-Hausdorff manifolds that are similar to "graphs with split vertices". It is shown that if $M$ is a connected one-dimensional non-Hausdorff manifold such that the set of its "non-Hausdorff" points is locally finite, and each component of its complement has a countable base, then there exists a quotient map $\ddot{I}€:M\to \hat{I}“$ onto an open one-dimensional CW complex, which maps the non-Hausdorff points of $M$ to the vertices of $\hat{I}“$. Moreover, $\hat{I}“$ is the minimal Hausdorff quotient of $M$, that is, for every continuous map $f:M\to N$ into a Hausdorff space $N$, there exists a unique continuous map $\hat{f}:\hat{I}“\to N$ such that $f=\hat{f}\circ \ddot{I}€$.

19 pages, 3 figures