Artículos científicos
Publicado 2019-02-19
Palabras clave
- Ideales en conjuntos numerables,
- Propiedades tipo Ramsey,
- p-ideales,
- p -ideales,
- q -ideales
- representación de ideales ...Más
Cómo citar
Uzcátegui Aylwin, C. (2019). Ideales sobre conjuntos numerables: un revisión con preguntas. Revista Integración, Temas De matemáticas, 37(1), 167–198. https://doi.org/10.18273/revint.v37n1-2019009
Resumen
Un ideal sobre un conjunto X es una colección de subconjuntos de X cerrada bajo las operaciones de tomar uniones finitas y subconjuntos de sus elementos. Los ideales son una noción muy útil en topología y teoría
de conjuntos y han sido estudiados desde hace mucho tiempo. Presentamos una revisión de algunos resultados sobre ideales en conjuntos numerables incluyendo preguntas abiertas sobre este tema.
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Referencias
[1] Barbarski P., Filipów R., Mrożek N. and Szuca P., “When does the Katětov order imply that one ideal extends the other?”, Colloq. Math. 130 (2013), No. 1, 91–102.
[2] Borodulin-Nadzieja P., Farkas B. and Plebanek G., “Representations of ideals in Polish groups and in Banach spaces”, J. Symb. Log. 80 (2015), No. 4, 1268–1289.
[3] Camargo J. and Uzcátegui C., “Selective separability on spaces with an analytic topology”, Topology Appl. 248 (2018), 176–191.
[4] Camargo J. and Uzcátegui C., “Some topological and combinatorial properties preserved by inverse limits”, Math. Slovaca 69 (2019), No. 1, 171–184.
[5] Cameron P.J., “The random graph”, in The mathematics of Paul Erdös, II, volume 14 of Algorithms Combin., Springer (1997), 333–351.
[6] Ciesielski K. and Jasinski J., “Topologies making a given ideal nowhere dense or meager”, Topology Appl. 63 (1995), 277–298.
[7] Debs G., “Effective properties in compact sets of Borel functions”, Mathematica 34 (1987), 64–68.
[8] Debs G., “Borel extractions of converging sequences in compact sets of Borel functions”, J. Math. Anal. Appl. 350 (2009), No. 2, 731–744.
[9] Dodos P., “Codings of separable compact subsets of the first Baire class”, Ann. Pure Appl. Logic 142 (2006), 425–441.
[10] Dodos P. and Kanellopoulos V., “On pairs of definable orthogonal families”, Illinois J. Math. 52 (2008), No. 1, 181–201.
[11] Farah I., “Ideals induced by Tsirelson submeasures”, Fund. Math. 159 (1999), No. 3, 243–258.
[12] Farah I., Analytic Quotiens, volume 148 of Mem. Amer. Math. Soc., Providence, Rhode Island, 2000.
[13] Farah I., “How many boolean algebras P(N)/I are there?”, Illinois J. Math. 46 (2003), 999–1033.
[14] Farah I., “Luzin gaps”, Trans. Amer. Math. Soc. 356 (2004), No. 6, 2197–2239.
[15] Farah I. and Solecki S., “Two F ideals”, Proc. Amer. Math. Soc. 131 (2003), No. 6, 1971–1975.
[16] Farkas B., Khomskii Y. and Vidnyánszky Z., “Almost disjoint refinements and mixing reals”, Fund. Math. 242 (2018), No. 1, 25–48.
[17] Filipów R., Mrożek N., Reclaw I. and Szuca P., “I-selection principles for sequences of functions”, J. Math. Anal. Appl. 396 (2012), No. 2, 680–688.
[18] Filipów R. and Szuca P., “Rearrangement of conditionally convergent series on a small set”, J. Math. Anal. Appl. 362 (2010), No. 1, 64–71.
[19] Fillipów R., Mrozek N., Reclaw I. and Szuca P., “Ideal convergence of bounded sequences”, J. symbolic Logic 72 (2007), No. 2, 501–512.
[20] Fillipów R., Mrozek N., Reclaw I. and Szuca P., “Ideal version of Ramsey’s theorem”, Czechoslovak Math. J. 61 (2011), No. 2, 289–308.
[21] García-Ferreira S. and Rivera-Gómez J.E., “Comparing Fréchet-Urysohn filters with two pre-orders”, Topology Appl. 225 (2017), 90–102.
[22] García-Ferreira S. and Rivera-Gómez J.E., “Ordering Fréchet-Urysohn filters”, Topology Appl. 163 (2014), 128 – 141.
[23] Grebík J. and Hru˘sák M., “No minimal tall Borel ideal in the Katětov order”, preprint, 2018.
[24] Grebík J. and Uzcátegui C., “Bases and Borel selectors for tall families”, J. Symb. Log., to appear, 2018.
[25] Guevara F., “Espacios uniformemente Fréchet”, Master’s thesis, Universidad de Los Andes, Mérida, Venezuela, 2011, 70 p.
[26] Guevara F., Personal communication, January, 2019.
[27] Guevara F. and Uzcátegui C., “Fréchet Borel ideals with Borel orthogonal”, Colloq. Math. 152 (2018), No. 1, 141–163.
[28] Hrušák M., “Combinatorics of filters and ideals”, in Set theory and its applications, volume 533 of Contemp. Math., Amer. Math. Soc. (2011), 29–69.
[29] Hrušák M., “Katětov order on Borel ideals”, Arch. Math. Logic 56 (2017), No. 7-8, 831–847.
[30] Hru˘sák M. and Meza-Alcántara D., “Comparison game on Borel ideals”, Comment. Math. Univ. Carolin. 52 (2011), No. 2, 191–204.
[31] Hru˘sák M., Meza-Alcántara D., Thümmel E. and Uzcátegui C., “Ramsey type properties of ideals”, Ann. Pure Appl. Logic 168 (2017), No. 11, 2022–2049.
[32] Hrušák M. and Meza-Alcántara D., “Universal submeasures and ideals”, Questions Answers Gen. Topology 31 (2013), No. 2, 65–69.
[33] Kechris A.S., Classical Descriptive Set Theory, Springer-Verlag, New York, 1995.
[34] Klinga P. and Nowik A. “Extendability to summable ideals”, Acta Math. Hungar. 152 (2017), No. 1, 150–160.
[35] Krawczyk A., “On the Rosenthal compacta and analytic sets”, Proc. Amer. Math. Soc. 115 (1992), No. 4, 1095–1100.
[36] Kwela A. and Sabok M., “Topological representations”, J. Math. Anal. Appl. 422 (2015), No. 2, 1434–1446.
[37] Kwela A. and Zakrzewski P., “Combinatorics of ideals—selectivity versus density”, Comment. Math. Univ. Carolin. 58 (2017), No. 2, 261–266.
[38] Kwela M. and Nowik A., “Ideals of nowhere dense sets in some topologies on positive integers”, Topology Appl. 248 (2018), 149 – 163.
[39] Laflamme C., “Filter games and combinatorial properties of strategies”, in Tomek (ed.) Bartoszynski, editor, Annual Boise extravaganza in set theory (BEST) conference,1992/1994, volume 192 of Contemp. Math., pages 51–67. American Mathematical Society., 1996.
[40] Laflamme C. and Leary C., “Filter games on ! and the dual ideal”, Fund. Math. 173 (2002), No. 2, 159–173.
[41] Mathias A.R.D., “Happy families”, Ann. Math. Log. 12 (1977), No. 1, 59–111.
[42] Mazur K., “F_sigma-ideals and w w*-gaps”, Fund. Math. 138 (1991), No. 2, 103–111.
[43] Meza-Alcántara D., “Ideals and filters on countable sets”, Thesis (Ph.D.), Universidad Nacional Autónoma de México, México, 2009, 139 p.
[44] Sabok M., Zapletal J, “Forcing properties of ideals of closed sets”, J. Symb. Log. 76 (2011), No. 3, 1075–1095.
[45] Sakai H., “On Katětov and Katětov-Blass orders on analytic P-ideals and Borel ideals”, Arch. Math. Logic 57 (2018), No. 3-4, 317–327.
[46] Samet N. and Tsaban B., “Superfilters, Ramsey theory, and van der Waerden’s Theorem”, Topology Appl., 156 (2009), 2659–2669.
[47] Shibakov A., “On sequential analytic groups”, Proc. Amer. Math. Soc. 145 (2017), No. 9, 4087–4096.
[48] Simon P., “A hedgehog in a product”, Acta. Univ. Carolin. Math. Phys. 39 (1998), 147–153.
[49] Solecki S., “Analytic ideals and their applications”, Ann. Pure Appl. Logic 99 (1999), No. 1-3, 51–72.
[50] Solecki S., “Filters and sequences”, Fund. Math., 163 (2000), 215–228.
[51] Solecki S., “Tukey reduction among analytic directed orders”, Zb. Rad. (Beogr.) 17 (2015), No. 25, (Selected topics in combinatorial analysis), 209–220.
[52] Solecki S. and Todorcevic S., “Avoiding families and Tukey functions on the nowhere-dense ideal”, J. Inst. Math. Jussieu 10 (2011), No. 2, 405–435.
[53] Solecki S. and Todorčević S., “Cofinal types of topological directed orders”, Annales de l’institut Fourier, 54 (2004), No. 6, 1877–1911.
[54] Todorčević S., “Analytic gaps”, Fund. Math. 150 (1996), No.1, 55–66.
[55] Todorčević S., Topics in Topology, Lecture Notes in Mathematics 1652, Springer, Berlin, 1997.
[56] Todorčević S., “Compacts sets of the first Baire class”, Journal of the AMS 12 (1999), No. 4, 1179–1212.
[57] Todorčević S., Introduction to Ramsey spaces, Annals of Mathematical Studies 174, Princeton University Press, 2010.
[58] Todorčević S. and Uzcátegui C., “Analytic topologies over countable sets”, Topology Appl. 111 (2001), No. 3, 299–326.
[59] Todorčević S. and Uzcátegui C., “Analytic k-spaces”, Topology Appl. 146-147 (2005), No. 1, 511–526.
[60] Todorčević S. and Uzcátegui C., “A nodec regular analytic space”, Topology Appl. 166 (2014), 85–91.
[61] Uzcátegui C., “On the complexity of the subspaces of S!”, Fund. Math. 176 (2003), 1–16.
[62] Uzcátegui C. and Vielma J. “Alexandroff topologies viewed as closed sets in the Cantor cube”, Divulg. Mat. 13 (2005), No. 1, 45–53.
[63] Veličković B., “A note on Tsirelson type ideals”, Fund. Math. 159 (1999), No. 3, 259–268.
[64] Zafrany S., “On analytic filters and prefilters”, J. Symb. Log. 55 (1990), No. 1, 315–322.
[2] Borodulin-Nadzieja P., Farkas B. and Plebanek G., “Representations of ideals in Polish groups and in Banach spaces”, J. Symb. Log. 80 (2015), No. 4, 1268–1289.
[3] Camargo J. and Uzcátegui C., “Selective separability on spaces with an analytic topology”, Topology Appl. 248 (2018), 176–191.
[4] Camargo J. and Uzcátegui C., “Some topological and combinatorial properties preserved by inverse limits”, Math. Slovaca 69 (2019), No. 1, 171–184.
[5] Cameron P.J., “The random graph”, in The mathematics of Paul Erdös, II, volume 14 of Algorithms Combin., Springer (1997), 333–351.
[6] Ciesielski K. and Jasinski J., “Topologies making a given ideal nowhere dense or meager”, Topology Appl. 63 (1995), 277–298.
[7] Debs G., “Effective properties in compact sets of Borel functions”, Mathematica 34 (1987), 64–68.
[8] Debs G., “Borel extractions of converging sequences in compact sets of Borel functions”, J. Math. Anal. Appl. 350 (2009), No. 2, 731–744.
[9] Dodos P., “Codings of separable compact subsets of the first Baire class”, Ann. Pure Appl. Logic 142 (2006), 425–441.
[10] Dodos P. and Kanellopoulos V., “On pairs of definable orthogonal families”, Illinois J. Math. 52 (2008), No. 1, 181–201.
[11] Farah I., “Ideals induced by Tsirelson submeasures”, Fund. Math. 159 (1999), No. 3, 243–258.
[12] Farah I., Analytic Quotiens, volume 148 of Mem. Amer. Math. Soc., Providence, Rhode Island, 2000.
[13] Farah I., “How many boolean algebras P(N)/I are there?”, Illinois J. Math. 46 (2003), 999–1033.
[14] Farah I., “Luzin gaps”, Trans. Amer. Math. Soc. 356 (2004), No. 6, 2197–2239.
[15] Farah I. and Solecki S., “Two F ideals”, Proc. Amer. Math. Soc. 131 (2003), No. 6, 1971–1975.
[16] Farkas B., Khomskii Y. and Vidnyánszky Z., “Almost disjoint refinements and mixing reals”, Fund. Math. 242 (2018), No. 1, 25–48.
[17] Filipów R., Mrożek N., Reclaw I. and Szuca P., “I-selection principles for sequences of functions”, J. Math. Anal. Appl. 396 (2012), No. 2, 680–688.
[18] Filipów R. and Szuca P., “Rearrangement of conditionally convergent series on a small set”, J. Math. Anal. Appl. 362 (2010), No. 1, 64–71.
[19] Fillipów R., Mrozek N., Reclaw I. and Szuca P., “Ideal convergence of bounded sequences”, J. symbolic Logic 72 (2007), No. 2, 501–512.
[20] Fillipów R., Mrozek N., Reclaw I. and Szuca P., “Ideal version of Ramsey’s theorem”, Czechoslovak Math. J. 61 (2011), No. 2, 289–308.
[21] García-Ferreira S. and Rivera-Gómez J.E., “Comparing Fréchet-Urysohn filters with two pre-orders”, Topology Appl. 225 (2017), 90–102.
[22] García-Ferreira S. and Rivera-Gómez J.E., “Ordering Fréchet-Urysohn filters”, Topology Appl. 163 (2014), 128 – 141.
[23] Grebík J. and Hru˘sák M., “No minimal tall Borel ideal in the Katětov order”, preprint, 2018.
[24] Grebík J. and Uzcátegui C., “Bases and Borel selectors for tall families”, J. Symb. Log., to appear, 2018.
[25] Guevara F., “Espacios uniformemente Fréchet”, Master’s thesis, Universidad de Los Andes, Mérida, Venezuela, 2011, 70 p.
[26] Guevara F., Personal communication, January, 2019.
[27] Guevara F. and Uzcátegui C., “Fréchet Borel ideals with Borel orthogonal”, Colloq. Math. 152 (2018), No. 1, 141–163.
[28] Hrušák M., “Combinatorics of filters and ideals”, in Set theory and its applications, volume 533 of Contemp. Math., Amer. Math. Soc. (2011), 29–69.
[29] Hrušák M., “Katětov order on Borel ideals”, Arch. Math. Logic 56 (2017), No. 7-8, 831–847.
[30] Hru˘sák M. and Meza-Alcántara D., “Comparison game on Borel ideals”, Comment. Math. Univ. Carolin. 52 (2011), No. 2, 191–204.
[31] Hru˘sák M., Meza-Alcántara D., Thümmel E. and Uzcátegui C., “Ramsey type properties of ideals”, Ann. Pure Appl. Logic 168 (2017), No. 11, 2022–2049.
[32] Hrušák M. and Meza-Alcántara D., “Universal submeasures and ideals”, Questions Answers Gen. Topology 31 (2013), No. 2, 65–69.
[33] Kechris A.S., Classical Descriptive Set Theory, Springer-Verlag, New York, 1995.
[34] Klinga P. and Nowik A. “Extendability to summable ideals”, Acta Math. Hungar. 152 (2017), No. 1, 150–160.
[35] Krawczyk A., “On the Rosenthal compacta and analytic sets”, Proc. Amer. Math. Soc. 115 (1992), No. 4, 1095–1100.
[36] Kwela A. and Sabok M., “Topological representations”, J. Math. Anal. Appl. 422 (2015), No. 2, 1434–1446.
[37] Kwela A. and Zakrzewski P., “Combinatorics of ideals—selectivity versus density”, Comment. Math. Univ. Carolin. 58 (2017), No. 2, 261–266.
[38] Kwela M. and Nowik A., “Ideals of nowhere dense sets in some topologies on positive integers”, Topology Appl. 248 (2018), 149 – 163.
[39] Laflamme C., “Filter games and combinatorial properties of strategies”, in Tomek (ed.) Bartoszynski, editor, Annual Boise extravaganza in set theory (BEST) conference,1992/1994, volume 192 of Contemp. Math., pages 51–67. American Mathematical Society., 1996.
[40] Laflamme C. and Leary C., “Filter games on ! and the dual ideal”, Fund. Math. 173 (2002), No. 2, 159–173.
[41] Mathias A.R.D., “Happy families”, Ann. Math. Log. 12 (1977), No. 1, 59–111.
[42] Mazur K., “F_sigma-ideals and w w*-gaps”, Fund. Math. 138 (1991), No. 2, 103–111.
[43] Meza-Alcántara D., “Ideals and filters on countable sets”, Thesis (Ph.D.), Universidad Nacional Autónoma de México, México, 2009, 139 p.
[44] Sabok M., Zapletal J, “Forcing properties of ideals of closed sets”, J. Symb. Log. 76 (2011), No. 3, 1075–1095.
[45] Sakai H., “On Katětov and Katětov-Blass orders on analytic P-ideals and Borel ideals”, Arch. Math. Logic 57 (2018), No. 3-4, 317–327.
[46] Samet N. and Tsaban B., “Superfilters, Ramsey theory, and van der Waerden’s Theorem”, Topology Appl., 156 (2009), 2659–2669.
[47] Shibakov A., “On sequential analytic groups”, Proc. Amer. Math. Soc. 145 (2017), No. 9, 4087–4096.
[48] Simon P., “A hedgehog in a product”, Acta. Univ. Carolin. Math. Phys. 39 (1998), 147–153.
[49] Solecki S., “Analytic ideals and their applications”, Ann. Pure Appl. Logic 99 (1999), No. 1-3, 51–72.
[50] Solecki S., “Filters and sequences”, Fund. Math., 163 (2000), 215–228.
[51] Solecki S., “Tukey reduction among analytic directed orders”, Zb. Rad. (Beogr.) 17 (2015), No. 25, (Selected topics in combinatorial analysis), 209–220.
[52] Solecki S. and Todorcevic S., “Avoiding families and Tukey functions on the nowhere-dense ideal”, J. Inst. Math. Jussieu 10 (2011), No. 2, 405–435.
[53] Solecki S. and Todorčević S., “Cofinal types of topological directed orders”, Annales de l’institut Fourier, 54 (2004), No. 6, 1877–1911.
[54] Todorčević S., “Analytic gaps”, Fund. Math. 150 (1996), No.1, 55–66.
[55] Todorčević S., Topics in Topology, Lecture Notes in Mathematics 1652, Springer, Berlin, 1997.
[56] Todorčević S., “Compacts sets of the first Baire class”, Journal of the AMS 12 (1999), No. 4, 1179–1212.
[57] Todorčević S., Introduction to Ramsey spaces, Annals of Mathematical Studies 174, Princeton University Press, 2010.
[58] Todorčević S. and Uzcátegui C., “Analytic topologies over countable sets”, Topology Appl. 111 (2001), No. 3, 299–326.
[59] Todorčević S. and Uzcátegui C., “Analytic k-spaces”, Topology Appl. 146-147 (2005), No. 1, 511–526.
[60] Todorčević S. and Uzcátegui C., “A nodec regular analytic space”, Topology Appl. 166 (2014), 85–91.
[61] Uzcátegui C., “On the complexity of the subspaces of S!”, Fund. Math. 176 (2003), 1–16.
[62] Uzcátegui C. and Vielma J. “Alexandroff topologies viewed as closed sets in the Cantor cube”, Divulg. Mat. 13 (2005), No. 1, 45–53.
[63] Veličković B., “A note on Tsirelson type ideals”, Fund. Math. 159 (1999), No. 3, 259–268.
[64] Zafrany S., “On analytic filters and prefilters”, J. Symb. Log. 55 (1990), No. 1, 315–322.