Artículo Original
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.
Descargas
Los datos de descargas todavía no están disponibles.
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.