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| Formato: | Preprint |
| Publicado: |
2026
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| Acceso en línea: | https://arxiv.org/abs/2604.16660 |
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| _version_ | 1866914485542846464 |
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| author | Grant, Benjamin |
| author_facet | Grant, Benjamin |
| contents | We define several topological spaces whose points are quivers with a given infinite vertex set $X$. In the special case when $X$ is countably infinite, we show that two of the spaces of interest are homeomorphic to the Baire space $\mathbb{N}^\mathbb{N}$. We study properties of countably infinite quivers as subspaces of these topological spaces and prove a ``meta-theorem'' about hereditary properties of quivers. Furthermore, we approach the question of convergence for infinite mutation sequences in these spaces, providing a complete characterization of the (non-)density of the domains of convergence and divergence of infinite mutation sequences in one of these spaces and a partial characterization in the other. We then draw attention to a very special infinite quiver which we call the \emph{Fraïssé quiver} that draws a clear contrast between the behavior of finite and infinite mutation sequences. Finally, we reproduce (a very mild modification of) a previously-constructed topological space due to Ervin and Jackson as a subquotient of one of the spaces of interest. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_16660 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Topologizing infinite quivers and their mutations Grant, Benjamin Combinatorics General Topology Logic 05E18, 13F60, 22F05, 54H05 We define several topological spaces whose points are quivers with a given infinite vertex set $X$. In the special case when $X$ is countably infinite, we show that two of the spaces of interest are homeomorphic to the Baire space $\mathbb{N}^\mathbb{N}$. We study properties of countably infinite quivers as subspaces of these topological spaces and prove a ``meta-theorem'' about hereditary properties of quivers. Furthermore, we approach the question of convergence for infinite mutation sequences in these spaces, providing a complete characterization of the (non-)density of the domains of convergence and divergence of infinite mutation sequences in one of these spaces and a partial characterization in the other. We then draw attention to a very special infinite quiver which we call the \emph{Fraïssé quiver} that draws a clear contrast between the behavior of finite and infinite mutation sequences. Finally, we reproduce (a very mild modification of) a previously-constructed topological space due to Ervin and Jackson as a subquotient of one of the spaces of interest. |
| title | Topologizing infinite quivers and their mutations |
| topic | Combinatorics General Topology Logic 05E18, 13F60, 22F05, 54H05 |
| url | https://arxiv.org/abs/2604.16660 |