Between closed and Ig-closed sets
The concept of closed sets is a central object in general topology. In order to extend many of important properties of closed sets to a larger families, Norman Levine initiated the study of generalized closed sets. In this paper we introduce, via ideals, new generalizations of closed subsets, which...
- Autores:
-
Pachon Rubiano, Néstor Raúl
- Tipo de recurso:
- Article of investigation
- Fecha de publicación:
- 2018
- Institución:
- Escuela Colombiana de Ingeniería Julio Garavito
- Repositorio:
- Repositorio Institucional ECI
- Idioma:
- eng
- OAI Identifier:
- oai:repositorio.escuelaing.edu.co:001/1393
- Acceso en línea:
- https://repositorio.escuelaing.edu.co/handle/001/1393
- Palabra clave:
- Mateméticas
Acciones de grupos (Matemáticas)
g-closed
Ig-closed
I-compact
I-normal
I-QHC
ρC(I)-compact.
g-closed
Ig-closed
I-compact
I-normal
I-QHC
ρC(I)-compact.
- Rights
- openAccess
- License
- https://creativecommons.org/licenses/by/4.0/
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| dc.title.spa.fl_str_mv |
Between closed and Ig-closed sets |
| title |
Between closed and Ig-closed sets |
| spellingShingle |
Between closed and Ig-closed sets Mateméticas Acciones de grupos (Matemáticas) g-closed Ig-closed I-compact I-normal I-QHC ρC(I)-compact. g-closed Ig-closed I-compact I-normal I-QHC ρC(I)-compact. |
| title_short |
Between closed and Ig-closed sets |
| title_full |
Between closed and Ig-closed sets |
| title_fullStr |
Between closed and Ig-closed sets |
| title_full_unstemmed |
Between closed and Ig-closed sets |
| title_sort |
Between closed and Ig-closed sets |
| dc.creator.fl_str_mv |
Pachon Rubiano, Néstor Raúl |
| dc.contributor.author.none.fl_str_mv |
Pachon Rubiano, Néstor Raúl |
| dc.contributor.researchgroup.spa.fl_str_mv |
Matemáticas |
| dc.subject.armarc.none.fl_str_mv |
Mateméticas Acciones de grupos (Matemáticas) |
| topic |
Mateméticas Acciones de grupos (Matemáticas) g-closed Ig-closed I-compact I-normal I-QHC ρC(I)-compact. g-closed Ig-closed I-compact I-normal I-QHC ρC(I)-compact. |
| dc.subject.proposal.spa.fl_str_mv |
g-closed Ig-closed I-compact I-normal I-QHC ρC(I)-compact. g-closed Ig-closed I-compact I-normal I-QHC ρC(I)-compact. |
| description |
The concept of closed sets is a central object in general topology. In order to extend many of important properties of closed sets to a larger families, Norman Levine initiated the study of generalized closed sets. In this paper we introduce, via ideals, new generalizations of closed subsets, which are strong forms of the Ig-closed sets, called ρIg-closed sets and closed-I sets. We present some properties and applications of these new sets and compare the ρIg-closed sets and the closed-I sets with the g-closed sets introduced by Levine. We show that I-closed and closed-I are independent concepts, as well as I * -closed sets and closed-I concepts. |
| publishDate |
2018 |
| dc.date.issued.none.fl_str_mv |
2018 |
| dc.date.accessioned.none.fl_str_mv |
2021-05-05T18:11:32Z 2021-10-01T17:20:51Z |
| dc.date.available.none.fl_str_mv |
2021-05-05 2021-10-01T17:20:51Z |
| dc.type.spa.fl_str_mv |
Artículo de revista |
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http://purl.org/coar/version/c_970fb48d4fbd8a85 |
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http://purl.org/coar/resource_type/c_2df8fbb1 |
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Text |
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http://purl.org/redcol/resource_type/ART |
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1307-5543 |
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https://repositorio.escuelaing.edu.co/handle/001/1393 |
| dc.identifier.doi.none.fl_str_mv |
10.29020/nybg.ejpam.v11i2.3131 |
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doi.org/10.29020/nybg.ejpam.v11i2.3131 |
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1307-5543 10.29020/nybg.ejpam.v11i2.3131 doi.org/10.29020/nybg.ejpam.v11i2.3131 |
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| dc.language.iso.spa.fl_str_mv |
eng |
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eng |
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European Journal of Pure and Applied Mathematics, Vol. 11, No. 1, 2018, 299-314 |
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314 |
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1 |
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299 |
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11 |
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N/A |
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European Journal of Pure and Applied Mathematics |
| dc.relation.references.eng.fl_str_mv |
V. Renuka Devi and D. Sivaraj. A generalization of normal spaces. Archivum Mathematicum, 44:265–270, 2008. S. Jafari and N. Rajesh. Generalized closed sets with respect to an ideal. Eur. Jour. of Pure and App. Math, 4(2):147–151, 2011. D. Jancovic and T. R. Hamlett. New topologies from old via ideals. Amer. Math. Monthly, 97:295–310, 1990. D. Jancovic and T. R. Hamlett. Compatible extensions of ideals. Bollettino U. M. I., (7):453–465, 1992. N. Levine. Generalized closed sets in Topology. Rend. Circ. Mat. Palermo, 19(2):89– 96, 1970. A. S. Mashhour, M. E. Abd El-Monsef, and S. N. El-Deep. On precontinuous and weak precontinuous mappings. Proc. Math. and Phys. Soc. of Egypt, 53:47–53, 1982 Abd El Monsef, E. F. Lashien, and A. A. Nasef. On I-open sets and I-continuous functions. Kyungpook Math. Jour., 32(1):21–30, 1992. R. L. Newcomb. Topologies which are compact modulo an ideal. PhD engthesis, Univ. of Calif. at Santa Barbara. California, 1967. N. R. Pachón. New forms of strong compactness in terms of ideals. Int. Jour. of Pure and App. Math., 106(2):481–493, 2016. N. R. Pachón. ρC(I)-compact and ρI-QHC spaces. Int. Jour. of Pure and App. Math., 108(2):199–214, 2016. ] J. Porter and J. Thomas. On H-closed and minimal Hausdorff spaces. Trans. Amer. Math. Soc., 138:159–170, 1969. ] R. Vaidyanathaswamy. The localization theory in set-topology. Proc. Indian Acad. Sci., 20:51–61, 1945. G. Viglino. C-compact spaces. Duke Mathematical Journal, 36(4):761–764, 1969. Suppose that X = ∪ α∈Λ Wα, where Wα ∈ τ ⊕ I for each α ∈ Λ. For all α ∈ Λ, there exist Vα ∈ τ and a collection {Ij}j∈Λα of elements in I, such that Wα = Vα ∪ ∪ j∈Λα Ij . Hence X = ∪ α∈Λ Vα ∪ ∪ α∈Λ ∪ j∈Λα Ij . Then X\ ∪ α∈Λ Vα ∈ I⊛ and since (X, τ, I ⊛) is ρI ⊛-compact, there exists Λ0 ⊆ Λ, finite, with X\ ∪ α∈Λ0 Vα ∈ I⊛. This implies that X\ ∪ α∈Λ0 Wα ∈ I⊛. Suppose that X\ ∪ α∈Λ Vα ∈ I, where {Vα}α∈Λ is a collection of elements in τ . There exists I ∈ I such that X\ ∪ α∈Λ Vα = I, and so X = I∪ ∪ α∈Λ Vα. Given that (X, τ ⊕ I) is compact there exists Λ0 ⊆ Λ, finite, with X = I∪ ∪ α∈Λ0 Vα. Hence X\ ∪ α∈Λ0 Vα ⊆ I ∈ I and X\ ∪ α∈Λ0 Vα ∈ I. Suppose that F\ ∪ α∈Λ Vα ∈ I, where {Vα}α∈Λ is a collection of elements in τ and F is closed in (X, τ ). There exists J ∈ I with F\ ∪ α∈Λ Vα = J, and so F ⊆ J ∪ ∪ α∈Λ Vα. Given that ( X, τ ⊕ I ) is C-compact and F is closed in ( X, τ ⊕ I ) , there exists Λ0 ⊆ Λ, finite, with F ⊆ adhτ⊕I (J) ∪ ∪ α∈Λ0 adhτ⊕I (Vα) ⊆ J ∪ ∪ α∈Λ0 Vα. Hence F\ ∪ α∈Λ0 Vα ⊆ J ∈ I and F\ ∪ α∈Λ0 Vα ∈ I. Parts (2) and (5) have similar demonstrations. □ |
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Pachon Rubiano, Néstor Raúld4f3434d033e2adbaa8e0f46ee7c56db600Matemáticas2021-05-05T18:11:32Z2021-10-01T17:20:51Z2021-05-052021-10-01T17:20:51Z20181307-5543https://repositorio.escuelaing.edu.co/handle/001/139310.29020/nybg.ejpam.v11i2.3131doi.org/10.29020/nybg.ejpam.v11i2.3131The concept of closed sets is a central object in general topology. In order to extend many of important properties of closed sets to a larger families, Norman Levine initiated the study of generalized closed sets. In this paper we introduce, via ideals, new generalizations of closed subsets, which are strong forms of the Ig-closed sets, called ρIg-closed sets and closed-I sets. We present some properties and applications of these new sets and compare the ρIg-closed sets and the closed-I sets with the g-closed sets introduced by Levine. We show that I-closed and closed-I are independent concepts, as well as I * -closed sets and closed-I concepts.El concepto de conjunto cerrado es un objeto central en la topología general. Con el fin de extender muchas de las propiedades importantes de los conjuntos cerrados a familias más amplias, Norman Levine inició el estudio de los conjuntos cerrados generalizados. En este trabajo introducimos, a través de ideales, nuevas generalizaciones de subconjuntos cerrados, que son formas fuertes de conjuntos cerrados. que son formas fuertes de los conjuntos cerrados Ig, llamados conjuntos cerrados ρIg y conjuntos cerrados-I. En presentamos algunas propiedades y aplicaciones de estos nuevos conjuntos y comparamos los conjuntos ρIg-cerrados y los conjuntos cerrados-I con los conjuntos g-cerrados introducidos por Levine. Demostramos que I-cerrado y cerrado-I son conceptos independientes, al igual que los conjuntos I * -cerrados y los conceptos cerrados-I.Néstor Raúl Pachón Rubiano Departamento de Matemáticas, Escuela Colombiana de Ingeniería, Bogotá, Colombia. Departamento de Matemáticas, Universidad Nacional, Bogotá, Colombia.Email addresses: nestor.pachon@escuelaing.edu.co, nrpachonr@unal.edu.co (N.R. Pachón)16 páginasapplication/pdfengBusiness Global LLCNew Yorkhttps://creativecommons.org/licenses/by/4.0/info:eu-repo/semantics/openAccessAtribución 4.0 Internacional (CC BY 4.0)http://purl.org/coar/access_right/c_abf2https://www.ejpam.com/index.php/ejpam/article/view/3131/608Between closed and Ig-closed setsArtículo de revistainfo:eu-repo/semantics/publishedVersionhttp://purl.org/coar/resource_type/c_2df8fbb1Textinfo:eu-repo/semantics/articlehttp://purl.org/redcol/resource_type/ARThttp://purl.org/coar/version/c_970fb48d4fbd8a85European Journal of Pure and Applied Mathematics, Vol. 11, No. 1, 2018, 299-314314129911N/AEuropean Journal of Pure and Applied MathematicsV. Renuka Devi and D. Sivaraj. A generalization of normal spaces. Archivum Mathematicum, 44:265–270, 2008.S. Jafari and N. Rajesh. Generalized closed sets with respect to an ideal. Eur. Jour. of Pure and App. Math, 4(2):147–151, 2011.D. Jancovic and T. R. Hamlett. New topologies from old via ideals. Amer. Math. Monthly, 97:295–310, 1990.D. Jancovic and T. R. Hamlett. Compatible extensions of ideals. Bollettino U. M. I., (7):453–465, 1992.N. Levine. Generalized closed sets in Topology. Rend. Circ. Mat. Palermo, 19(2):89– 96, 1970.A. S. Mashhour, M. E. Abd El-Monsef, and S. N. El-Deep. On precontinuous and weak precontinuous mappings. Proc. Math. and Phys. Soc. of Egypt, 53:47–53, 1982Abd El Monsef, E. F. Lashien, and A. A. Nasef. On I-open sets and I-continuous functions. Kyungpook Math. Jour., 32(1):21–30, 1992.R. L. Newcomb. Topologies which are compact modulo an ideal. PhD engthesis, Univ. of Calif. at Santa Barbara. California, 1967.N. R. Pachón. New forms of strong compactness in terms of ideals. Int. Jour. of Pure and App. Math., 106(2):481–493, 2016.N. R. Pachón. ρC(I)-compact and ρI-QHC spaces. Int. Jour. of Pure and App. Math., 108(2):199–214, 2016.] J. Porter and J. Thomas. On H-closed and minimal Hausdorff spaces. Trans. Amer. Math. Soc., 138:159–170, 1969.] R. Vaidyanathaswamy. The localization theory in set-topology. Proc. Indian Acad. Sci., 20:51–61, 1945.G. Viglino. C-compact spaces. Duke Mathematical Journal, 36(4):761–764, 1969.Suppose that X = ∪ α∈Λ Wα, where Wα ∈ τ ⊕ I for each α ∈ Λ. For all α ∈ Λ, there exist Vα ∈ τ and a collection {Ij}j∈Λα of elements in I, such that Wα = Vα ∪ ∪ j∈Λα Ij . Hence X = ∪ α∈Λ Vα ∪ ∪ α∈Λ ∪ j∈Λα Ij . Then X\ ∪ α∈Λ Vα ∈ I⊛ and since (X, τ, I ⊛) is ρI ⊛-compact, there exists Λ0 ⊆ Λ, finite, with X\ ∪ α∈Λ0 Vα ∈ I⊛. This implies that X\ ∪ α∈Λ0 Wα ∈ I⊛.Suppose that X\ ∪ α∈Λ Vα ∈ I, where {Vα}α∈Λ is a collection of elements in τ . There exists I ∈ I such that X\ ∪ α∈Λ Vα = I, and so X = I∪ ∪ α∈Λ Vα. Given that (X, τ ⊕ I) is compact there exists Λ0 ⊆ Λ, finite, with X = I∪ ∪ α∈Λ0 Vα. Hence X\ ∪ α∈Λ0 Vα ⊆ I ∈ I and X\ ∪ α∈Λ0 Vα ∈ I.Suppose that F\ ∪ α∈Λ Vα ∈ I, where {Vα}α∈Λ is a collection of elements in τ and F is closed in (X, τ ). There exists J ∈ I with F\ ∪ α∈Λ Vα = J, and so F ⊆ J ∪ ∪ α∈Λ Vα. Given that ( X, τ ⊕ I ) is C-compact and F is closed in ( X, τ ⊕ I ) , there exists Λ0 ⊆ Λ, finite, with F ⊆ adhτ⊕I (J) ∪ ∪ α∈Λ0 adhτ⊕I (Vα) ⊆ J ∪ ∪ α∈Λ0 Vα. Hence F\ ∪ α∈Λ0 Vα ⊆ J ∈ I and F\ ∪ α∈Λ0 Vα ∈ I. Parts (2) and (5) have similar demonstrations. □MateméticasAcciones de grupos (Matemáticas)g-closedIg-closedI-compactI-normalI-QHCρC(I)-compact.g-closedIg-closedI-compactI-normalI-QHCρC(I)-compact.TEXTBetween closed and Ig-closed sets.pdf.txtBetween closed and Ig-closed sets.pdf.txtExtracted texttext/plain32379https://repositorio.escuelaing.edu.co/bitstream/001/1393/4/Between%20closed%20and%20Ig-closed%20sets.pdf.txt2215186f6caea92473f83dd9ed0d7095MD54open accessLICENSElicense.txttext/plain1881https://repositorio.escuelaing.edu.co/bitstream/001/1393/2/license.txt5a7ca94c2e5326ee169f979d71d0f06eMD52open accessORIGINALBetween closed and Ig-closed sets.pdfapplication/pdf246166https://repositorio.escuelaing.edu.co/bitstream/001/1393/3/Between%20closed%20and%20Ig-closed%20sets.pdfadd7fa11541d067fdb74eac8eb3e2cf3MD53open accessTHUMBNAILBetween closed and Ig-closed sets.pdf.jpgBetween closed and Ig-closed sets.pdf.jpgGenerated Thumbnailimage/jpeg10764https://repositorio.escuelaing.edu.co/bitstream/001/1393/5/Between%20closed%20and%20Ig-closed%20sets.pdf.jpg032840236f58be697ed9507dd1ea9a72MD55open access001/1393oai:repositorio.escuelaing.edu.co:001/13932021-10-01 17:35:58.547open accessRepositorio Escuela Colombiana de Ingeniería Julio Garavitorepositorio.eci@escuelaing.edu.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 |