2023/04/28 by Richard N. Ball, Ball, Richard N.
Mathematics · #06D22 (Primary) 54C45 #54G10 (Secondary) #54G12 #Advanced Topology and Set Theory #FOS: Mathematics #General Topology (math.GN) #Homotopy and Cohomology in Algebraic Topology #Rings, Modules, and Algebras
paper · pdf · doi:10.48550/arxiv.2305.00096
openalex publication_date 2023/04/28 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
(Completely regular) locales generalize (Tychonoff) spaces; indeed, the passage from a locale to its spatial sublocale is a well understood coreflection. But a locale also possesses an equally important pointless sublocale, and with morphisms suitably restricted, the passage from a locale to its pointless sublocale is also a coreflection. Our main theorem is that every locale can be uniquely represented as a subdirect product of its pointless and spatial parts, again with suitably restricted projections. We then exploit this representation by showing that any locale is determined by (what may be described as) the placement of its points in its pointless part.