Remainders in pointfree topology

• Autor(a) principal: Ferreira, Maria João
• Data de Publicação: 2018
• Outros Autores: Picado, Jorge, Marques Pinto, Sandra
• Tipo de documento: Artigo
• Idioma: eng
• Título da fonte: Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos)
• Texto Completo: http://hdl.handle.net/10316/90475; https://doi.org/10.1016/j.topol.2018.06.007
• topic: Frame; Locale; Sublocale; Heyting algebra; Coframe; Pseudodifference; Remainder; Remainder preservation; Proper map; Stone–Čech compactification; Regular Lindelöf reflection; Realcompact reflection; Nearly realcompact; Nearly pseudocompact; Hyper-real map

Remainders of subspaces are important e.g. in the realm of compactifications. Their extension to pointfree topology faces a difficulty: sublocale lattices are more complicated than their topological counterparts (complete atomic Boolean algebras). Nevertheless, the co-Heyting structure of sublocale lattices is enough to provide a counterpart to subspace remainders: the sublocale supplements. In this paper we give an account of their fundamental properties, emphasizing their similarities and differences with classical remainders, and provide several examples and applications to illustrate their scope. In particular, we study their behavior under image and preimage maps, as well as their preservation by pointfree continuous maps (i.e. localic maps). We then use them to characterize nearly realcompact and nearly pseudocompact frames. In addition, we introduce and study hyper-real localic maps.