Publicação
A pointfree theory of Pervin spaces
| Resumo: | We lay down the foundations for a pointfree theory of Pervin spaces. APervin space is a set equipped with a bounded sublattice of its powerset, and it isknown that these objects characterize those quasi-uniform spaces that are transitiveand totally bounded. The pointfree notion of a Pervin space, which we call Frithframe, consists of a frame equipped with a generating bounded sublattice. In thispaper we introduce and study the category of Frith frames and show that the classicaldual adjunction between topological spaces and frames extends to a dual adjunctionbetween Pervin spaces and Frith frames. Unlike what happens for Pervin spaces, wedo not have an equivalence between the categories of transitive and totally boundedquasi-uniform frames and of Frith frames, but we show that the latter is a fullcoreflective subcategory of the former. We also explore the notion of completenessof Frith frames inherited from quasi-uniform frames, providing a characterizationof those Frith frames that are complete and a description of the completion of anarbitrary Frith frame. |
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| Autores principais: | Borlido, Célia |
| Outros Autores: | Suarez, Anna Laura |
| Assunto: | Pervinspace Frithframe Firthquasi-uniformity transitive and totally bounded quasi-uniformity completion |
| Ano: | 2023 |
| País: | Portugal |
| Tipo de documento: | artigo |
| Tipo de acesso: | acesso aberto |
| Instituição associada: | Universidade de Coimbra |
| Idioma: | inglês |
| Origem: | Estudo Geral - Universidade de Coimbra |
| Resumo: | We lay down the foundations for a pointfree theory of Pervin spaces. APervin space is a set equipped with a bounded sublattice of its powerset, and it isknown that these objects characterize those quasi-uniform spaces that are transitiveand totally bounded. The pointfree notion of a Pervin space, which we call Frithframe, consists of a frame equipped with a generating bounded sublattice. In thispaper we introduce and study the category of Frith frames and show that the classicaldual adjunction between topological spaces and frames extends to a dual adjunctionbetween Pervin spaces and Frith frames. Unlike what happens for Pervin spaces, wedo not have an equivalence between the categories of transitive and totally boundedquasi-uniform frames and of Frith frames, but we show that the latter is a fullcoreflective subcategory of the former. We also explore the notion of completenessof Frith frames inherited from quasi-uniform frames, providing a characterizationof those Frith frames that are complete and a description of the completion of anarbitrary Frith frame. |
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