Scott locales
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2025
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| _version_ | 1866908710945685504 |
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| author | Resende, Pedro Santos, João Paulo |
| author_facet | Resende, Pedro Santos, João Paulo |
| contents | We prove some facts about locales $L$ equipped with the Scott topology $Ω(L)$, in particular studying a canonical frame homomorphism $ϕ:Ω(L)\to L$ which is motivated by an application to cognitive science. Such a topological locale $L$ is called a Scott locale if the inclusion of primes $p:Σ(L)\to L$ is continuous. We prove that the spectrum $Σ(L)$ of a Scott locale $L$ is necessarily $T_1$, and that preregular locales (a generalization of regular locales) are Scott locales. If $L$ is the topology of a topological space $X$ we find a (necessarily unique) continuous map $f:X\to L$ such that $f^{-1}=ϕ$ and compare it with the points-to-primes map $p:X\to L$, showing that $f=p$ if and only if $X$ is preregular, and that a sober space $X$ is Hausdorff if and only if $X$ is $T_1$ and $f(X)\subseteqΣ(L)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_14892 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Scott locales Resende, Pedro Santos, João Paulo Category Theory General Topology 18F70, 54D10, 54E99, 54H12 We prove some facts about locales $L$ equipped with the Scott topology $Ω(L)$, in particular studying a canonical frame homomorphism $ϕ:Ω(L)\to L$ which is motivated by an application to cognitive science. Such a topological locale $L$ is called a Scott locale if the inclusion of primes $p:Σ(L)\to L$ is continuous. We prove that the spectrum $Σ(L)$ of a Scott locale $L$ is necessarily $T_1$, and that preregular locales (a generalization of regular locales) are Scott locales. If $L$ is the topology of a topological space $X$ we find a (necessarily unique) continuous map $f:X\to L$ such that $f^{-1}=ϕ$ and compare it with the points-to-primes map $p:X\to L$, showing that $f=p$ if and only if $X$ is preregular, and that a sober space $X$ is Hausdorff if and only if $X$ is $T_1$ and $f(X)\subseteqΣ(L)$. |
| title | Scott locales |
| topic | Category Theory General Topology 18F70, 54D10, 54E99, 54H12 |
| url | https://arxiv.org/abs/2511.14892 |