One-dimensional non-Hausdorff manifolds and CW complexes

TL;DR

This paper characterizes certain one-dimensional non-Hausdorff manifolds by showing they can be quotiented onto minimal Hausdorff CW complexes, revealing their structure in relation to graphs with split vertices.

Contribution

It establishes a quotient map from connected one-dimensional non-Hausdorff manifolds to minimal Hausdorff CW complexes, generalizing the structure of graphs with split vertices.

Findings

Existence of a quotient map onto a minimal Hausdorff CW complex.

The CW complex is a minimal Hausdorff quotient of the manifold.

Non-Hausdorff points correspond to vertices in the CW complex.

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 $\pi\colon M \to \Gamma$ onto an open one-dimensional CW complex, which maps the non-Hausdorff points of $M$ to the vertices of $\Gamma$. Moreover, $\Gamma$ is the minimal Hausdorff quotient of $M$, that is, for every continuous map $f\colon M \to N$ into a Hausdorff space $N$, there exists a unique continuous map $\hat{f}\colon \Gamma \to N$ such that $f = \hat{f} \circ \pi$.