One-dimensional non-Hausdorff manifolds and CW complexes
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866910160651288576 |
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| author | Vlasenko, Igor Maksymenko, Sergiy |
| author_facet | Vlasenko, Igor Maksymenko, Sergiy |
| contents | 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 $π\colon M \to Γ$ onto an open one-dimensional CW complex, which maps the non-Hausdorff points of $M$ to the vertices of $Γ$.
Moreover, $Γ$ 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 $\hat{f}\colon Γ\to N$ such that $f = \hat{f} \circ π$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_21868 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | One-dimensional non-Hausdorff manifolds and CW complexes Vlasenko, Igor Maksymenko, Sergiy Geometric Topology Algebraic Topology Differential Geometry Dynamical Systems General Topology 57Q05, 54B15, 57N80 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 $π\colon M \to Γ$ onto an open one-dimensional CW complex, which maps the non-Hausdorff points of $M$ to the vertices of $Γ$. Moreover, $Γ$ 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 $\hat{f}\colon Γ\to N$ such that $f = \hat{f} \circ π$. |
| title | One-dimensional non-Hausdorff manifolds and CW complexes |
| topic | Geometric Topology Algebraic Topology Differential Geometry Dynamical Systems General Topology 57Q05, 54B15, 57N80 |
| url | https://arxiv.org/abs/2604.21868 |