Constructions of Lindelöf scattered P-spaces

We construct locally Lindelöf scattered P-spaces (LLSP spaces, for short) with prescribed widths and heights under different set-theoretic assumptions. We prove that there is an LLSP space of width $\omega_1$ and height $\omega_2$ and that it is relatively consistent with ZFC that there is an LLSP s...

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Authors: Martínez Alonso, Juan Carlos, Soukup, Lajos
Format: article
Status:Versión aceptada para publicación
Publication Date:2022
Country:España
Institution:Varias* (Consorci de Biblioteques Universitáries de Catalunya, Centre de Serveis Científics i Acadèmics de Catalunya)
Repository:Recercat. Dipósit de la Recerca de Catalunya
OAI Identifier:oai:recercat.cat:2445/194104
Online Access:https://hdl.handle.net/2445/194104
Access Level:Open access
Keyword:Nombres cardinals
Teoria de conjunts
Topologia
Espais topològics
Cardinal numbers
Set theory
Topology
Topological spaces
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repository_id_str
spelling Constructions of Lindelöf scattered P-spacesMartínez Alonso, Juan CarlosSoukup, LajosNombres cardinalsTeoria de conjuntsTopologiaEspais topològicsCardinal numbersSet theoryTopologyTopological spacesWe construct locally Lindelöf scattered P-spaces (LLSP spaces, for short) with prescribed widths and heights under different set-theoretic assumptions. We prove that there is an LLSP space of width $\omega_1$ and height $\omega_2$ and that it is relatively consistent with ZFC that there is an LLSP space of width $\omega_1$ and height $\omega_3$. Also, we prove a stepping up theorem which, for every cardinal $\lambda \geq \omega_2$, permits us to construct from an LLSP space of width $\omega_1$ and height $\lambda$ satisfying certain additional properties an LLSP space of width $\omega_1$ and height $\alpha$ for every ordinal $\alpha<\lambda^{+}$. As consequences of the above results, we obtain the following theorems: (1) For every ordinal $\alpha<\omega_3$ there is an LLSP space of width $\omega_1$ and height $\alpha$. (2) It is relatively consistent with ZFC that there is an LLSP space of width $\omega_1$ and height $\alpha$ for every ordinal $\alpha<\omega_4$.Institute of Mathematics, Polish Academy of Sciences2023202320222023info:eu-repo/semantics/articleinfo:eu-repo/semantics/acceptedVersion16 p.application/pdfhttps://hdl.handle.net/2445/194104Articles publicats en revistes (Matemàtiques i Informàtica)reponame:Recercat. Dipósit de la Recerca de Catalunyainstname:Varias* (Consorci de Biblioteques Universitáries de Catalunya, Centre de Serveis Científics i Acadèmics de Catalunya)InglésVersió postprint del document publicat a: https://doi.org/10.4064/fm228-7-2022Fundamenta Mathematicae, 2022, vol. 259, num. 3, p. 271-286https://doi.org/10.4064/fm228-7-2022(c) Institute of Mathematics, Polish Academy of Sciences, 2022info:eu-repo/semantics/openAccessoai:recercat.cat:2445/1941042026-05-29T05:05:01Z
dc.title.none.fl_str_mv Constructions of Lindelöf scattered P-spaces
title Constructions of Lindelöf scattered P-spaces
spellingShingle Constructions of Lindelöf scattered P-spaces
Martínez Alonso, Juan Carlos
Nombres cardinals
Teoria de conjunts
Topologia
Espais topològics
Cardinal numbers
Set theory
Topology
Topological spaces
title_short Constructions of Lindelöf scattered P-spaces
title_full Constructions of Lindelöf scattered P-spaces
title_fullStr Constructions of Lindelöf scattered P-spaces
title_full_unstemmed Constructions of Lindelöf scattered P-spaces
title_sort Constructions of Lindelöf scattered P-spaces
dc.creator.none.fl_str_mv Martínez Alonso, Juan Carlos
Soukup, Lajos
author Martínez Alonso, Juan Carlos
author_facet Martínez Alonso, Juan Carlos
Soukup, Lajos
author_role author
author2 Soukup, Lajos
author2_role author
dc.subject.none.fl_str_mv Nombres cardinals
Teoria de conjunts
Topologia
Espais topològics
Cardinal numbers
Set theory
Topology
Topological spaces
topic Nombres cardinals
Teoria de conjunts
Topologia
Espais topològics
Cardinal numbers
Set theory
Topology
Topological spaces
description We construct locally Lindelöf scattered P-spaces (LLSP spaces, for short) with prescribed widths and heights under different set-theoretic assumptions. We prove that there is an LLSP space of width $\omega_1$ and height $\omega_2$ and that it is relatively consistent with ZFC that there is an LLSP space of width $\omega_1$ and height $\omega_3$. Also, we prove a stepping up theorem which, for every cardinal $\lambda \geq \omega_2$, permits us to construct from an LLSP space of width $\omega_1$ and height $\lambda$ satisfying certain additional properties an LLSP space of width $\omega_1$ and height $\alpha$ for every ordinal $\alpha<\lambda^{+}$. As consequences of the above results, we obtain the following theorems: (1) For every ordinal $\alpha<\omega_3$ there is an LLSP space of width $\omega_1$ and height $\alpha$. (2) It is relatively consistent with ZFC that there is an LLSP space of width $\omega_1$ and height $\alpha$ for every ordinal $\alpha<\omega_4$.
publishDate 2022
dc.date.none.fl_str_mv 2022
2023
2023
2023
dc.type.none.fl_str_mv info:eu-repo/semantics/article
info:eu-repo/semantics/acceptedVersion
format article
status_str acceptedVersion
dc.identifier.none.fl_str_mv https://hdl.handle.net/2445/194104
url https://hdl.handle.net/2445/194104
dc.language.none.fl_str_mv Inglés
language_invalid_str_mv Inglés
dc.relation.none.fl_str_mv Versió postprint del document publicat a: https://doi.org/10.4064/fm228-7-2022
Fundamenta Mathematicae, 2022, vol. 259, num. 3, p. 271-286
https://doi.org/10.4064/fm228-7-2022
dc.rights.none.fl_str_mv (c) Institute of Mathematics, Polish Academy of Sciences, 2022
info:eu-repo/semantics/openAccess
rights_invalid_str_mv (c) Institute of Mathematics, Polish Academy of Sciences, 2022
eu_rights_str_mv openAccess
dc.format.none.fl_str_mv 16 p.
application/pdf
dc.publisher.none.fl_str_mv Institute of Mathematics, Polish Academy of Sciences
publisher.none.fl_str_mv Institute of Mathematics, Polish Academy of Sciences
dc.source.none.fl_str_mv Articles publicats en revistes (Matemàtiques i Informàtica)
reponame:Recercat. Dipósit de la Recerca de Catalunya
instname:Varias* (Consorci de Biblioteques Universitáries de Catalunya, Centre de Serveis Científics i Acadèmics de Catalunya)
instname_str Varias* (Consorci de Biblioteques Universitáries de Catalunya, Centre de Serveis Científics i Acadèmics de Catalunya)
reponame_str Recercat. Dipósit de la Recerca de Catalunya
collection Recercat. Dipósit de la Recerca de Catalunya
repository.name.fl_str_mv
repository.mail.fl_str_mv
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score 15.81155

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