Axiom TD and the Simmons sublocale theorem

Detalhes bibliográficos
Autor(a) principal: Picado, Jorge
Data de Publicação: 2019
Outros Autores: Pultr, Aleš
Tipo de documento: Artigo
Idioma: eng
Título da fonte: Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos)
Texto Completo: http://hdl.handle.net/10316/90469
https://doi.org/10.14712/1213-7243.2019.030
Resumo: More precisely, we are analyzing some of H. Simmons, S. B. Niefield and K. I. Rosenthal results concerning sublocales induced by subspaces. H. Simmons was concerned with the question when the coframe of sublocales is Boolean; he recognized the role of the axiom $T_D$ for the relation of certain degrees of scatteredness but did not emphasize its role in the relation {between} sublocales and subspaces. S. B. Niefield and K. I. Rosenthal just mention this axiom in a remark about Simmons' result. In this paper we show that the role of $T_D$ in this question is crucial. Concentration on the properties of $T_D$-spaces and technique of sublocales in this context allows us to present a simple, transparent and choice-free proof of the scatteredness theorem.
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spelling Axiom TD and the Simmons sublocale theoremFrame; locale; sublocale; coframe of sublocales; spatial sublocale; induced sublocale; $T_D$-separation; covered prime element; scattered space; weakly scattered space.More precisely, we are analyzing some of H. Simmons, S. B. Niefield and K. I. Rosenthal results concerning sublocales induced by subspaces. H. Simmons was concerned with the question when the coframe of sublocales is Boolean; he recognized the role of the axiom $T_D$ for the relation of certain degrees of scatteredness but did not emphasize its role in the relation {between} sublocales and subspaces. S. B. Niefield and K. I. Rosenthal just mention this axiom in a remark about Simmons' result. In this paper we show that the role of $T_D$ in this question is crucial. Concentration on the properties of $T_D$-spaces and technique of sublocales in this context allows us to present a simple, transparent and choice-free proof of the scatteredness theorem.Mathematical Institute of Charles University2019info:eu-repo/semantics/publishedVersioninfo:eu-repo/semantics/articlehttp://hdl.handle.net/10316/90469http://hdl.handle.net/10316/90469https://doi.org/10.14712/1213-7243.2019.030enghttps://cmuc.karlin.mff.cuni.cz/cmuc1904/abs/picpul.htmPicado, JorgePultr, Alešinfo:eu-repo/semantics/openAccessreponame:Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos)instname:Agência para a Sociedade do Conhecimento (UMIC) - FCT - Sociedade da Informaçãoinstacron:RCAAP2022-05-25T03:12:30Zoai:estudogeral.uc.pt:10316/90469Portal AgregadorONGhttps://www.rcaap.pt/oai/openaireopendoar:71602024-03-19T21:10:35.992016Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos) - Agência para a Sociedade do Conhecimento (UMIC) - FCT - Sociedade da Informaçãofalse
dc.title.none.fl_str_mv Axiom TD and the Simmons sublocale theorem
title Axiom TD and the Simmons sublocale theorem
spellingShingle Axiom TD and the Simmons sublocale theorem
Picado, Jorge
Frame; locale; sublocale; coframe of sublocales; spatial sublocale; induced sublocale; $T_D$-separation; covered prime element; scattered space; weakly scattered space.
title_short Axiom TD and the Simmons sublocale theorem
title_full Axiom TD and the Simmons sublocale theorem
title_fullStr Axiom TD and the Simmons sublocale theorem
title_full_unstemmed Axiom TD and the Simmons sublocale theorem
title_sort Axiom TD and the Simmons sublocale theorem
author Picado, Jorge
author_facet Picado, Jorge
Pultr, Aleš
author_role author
author2 Pultr, Aleš
author2_role author
dc.contributor.author.fl_str_mv Picado, Jorge
Pultr, Aleš
dc.subject.por.fl_str_mv Frame; locale; sublocale; coframe of sublocales; spatial sublocale; induced sublocale; $T_D$-separation; covered prime element; scattered space; weakly scattered space.
topic Frame; locale; sublocale; coframe of sublocales; spatial sublocale; induced sublocale; $T_D$-separation; covered prime element; scattered space; weakly scattered space.
description More precisely, we are analyzing some of H. Simmons, S. B. Niefield and K. I. Rosenthal results concerning sublocales induced by subspaces. H. Simmons was concerned with the question when the coframe of sublocales is Boolean; he recognized the role of the axiom $T_D$ for the relation of certain degrees of scatteredness but did not emphasize its role in the relation {between} sublocales and subspaces. S. B. Niefield and K. I. Rosenthal just mention this axiom in a remark about Simmons' result. In this paper we show that the role of $T_D$ in this question is crucial. Concentration on the properties of $T_D$-spaces and technique of sublocales in this context allows us to present a simple, transparent and choice-free proof of the scatteredness theorem.
publishDate 2019
dc.date.none.fl_str_mv 2019
dc.type.status.fl_str_mv info:eu-repo/semantics/publishedVersion
dc.type.driver.fl_str_mv info:eu-repo/semantics/article
format article
status_str publishedVersion
dc.identifier.uri.fl_str_mv http://hdl.handle.net/10316/90469
http://hdl.handle.net/10316/90469
https://doi.org/10.14712/1213-7243.2019.030
url http://hdl.handle.net/10316/90469
https://doi.org/10.14712/1213-7243.2019.030
dc.language.iso.fl_str_mv eng
language eng
dc.relation.none.fl_str_mv https://cmuc.karlin.mff.cuni.cz/cmuc1904/abs/picpul.htm
dc.rights.driver.fl_str_mv info:eu-repo/semantics/openAccess
eu_rights_str_mv openAccess
dc.publisher.none.fl_str_mv Mathematical Institute of Charles University
publisher.none.fl_str_mv Mathematical Institute of Charles University
dc.source.none.fl_str_mv reponame:Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos)
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instname_str Agência para a Sociedade do Conhecimento (UMIC) - FCT - Sociedade da Informação
instacron_str RCAAP
institution RCAAP
reponame_str Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos)
collection Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos)
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