Ir al contenido

Documat

On traditional Menger and Rothberger variations

  • Autores: Christopher Caruvana, Steven Clontz, Jared Holshouser
  • Localización: Applied general topology, ISSN-e 1989-4147, ISSN 1576-9402, Vol. 25, Nº. 2, 2024, págs. 519-552
  • Idioma: inglés
  • DOI: 10.4995/agt.2024.21437
  • Enlaces
  • Resumen

    • We present a comprehensive report on the relationships between variations of the Menger and Rothberger selection properties with respect to w-covers and k-covers in the most general topological setting and address the finite productivity of some of these properties. We collect various examples that separate certain properties and we carefully identify which separation axioms simplify aspects of these properties. We finish with a consolidated list of open questions focused on topological examples.

  • Referencias bibliográficas
    • A. V. Arhangel'skii, From classic topological invariants to relative topological properties, Sci. Math. Jpn. 55, no. 1 (2001), 153-201.
    • C. Caruvana and J. Holshouser, Closed discrete selection in the compact open topology, Topology Proc. 56 (2020), 25-55.
    • C. Caruvana and J. Holshouser, Selection games and the Vietoris space, Topology Appl. 307 (2022), 107772. https://doi.org/10.1016/j.topol.2021.107772
    • C. Caruvana and J. Holshouser, Translation results for some star-selection games, Topology Appl. 348 (2024), 108889. https://doi.org/10.1016/j.topol.2024.108889
    • C. Caruvana and R. R. Kallman, An extension of the Baire property, Topology Proc. 55 (2020), 87-114.
    • A. Caserta, G. Di Maio, Lj. D. R. Kočinac, and E. Meccariello, Applications of k-covers II, Topology Appl. 153, no. 17 (2006), 3277-3293....
    • S. Clontz, Applications of limited information strategies in Menger's game, Commentat. Math. Univ. Carol. 58, no. 2 (2017), 225-239. https://doi.org/10.14712/1213-7243.2015.201
    • S. Clontz, Relating games of Menger, countable fan tightness, and selective separability, Topology Appl. 258 (2019), 215-228. https://doi.org/10.1016/j.topol.2019.02.062
    • S. Clontz, Dual selection games, Topology Appl. 272 (2020), 107056. https://doi.org/10.1016/j.topol.2020.107056
    • S. Clontz and J. Holshouser, Limited information strategies and discrete selectivity, Topology Appl. 265 (2019), 106815. https://doi.org/10.1016/j.topol.2019.07.008
    • C. Costantini, L. Holá, and P. Vitolo, Tightness, character and related properties of hyperspace topologies, Topology Appl. 142, no. 1-3 (2004),...
    • M. Davis, Infinite games of perfect information, Ann. Math. Stud. 52 (1964), 85-101. https://doi.org/10.1515/9781400882014-008
    • G. Di Maio, L. D. R. Kočinac, and E. Meccariello, Applications of k-covers, Acta Math. Sin., Engl. Ser. 22, no. 4 (2006), 1151-1160. https://doi.org/10.1007/s10114-005-0753-8
    • R. R. Dias and M. Scheepers, Selective games on binary relations, Topology Appl. 192 (2015), 58-83. https://doi.org/10.1016/j.topol.2015.05.071
    • R. Engelking, General Topology, Sigma series in pure mathematics, vol. 6, Heldermann Verlag, 1989.
    • D. Gale and F. M. Stewart, Infinite games with perfect information, Contrib. Theory of Games, II, Ann. Math. Stud. No. 28, 1953, pp. 245-266....
    • F. Galvin, Indeterminacy of point-open games, Bull. Acad. Pol. Sci. Sér. Sci. Math. Astron. Phys. 26 (1978), 445-449.
    • F. Galvin and R. Telgársky, Stationary strategies in topological games, Topology Appl. 22 (1986), 51-69. https://doi.org/10.1016/0166-8641(86)90077-5
    • J. Gerlits and Zs. Nagy, Some properties of C(X). I, Topology Appl. 14 (1982), 151-161. https://doi.org/10.1016/0166-8641(82)90065-7
    • W. Hurewicz, Über eine Verallgemeinerung des Borelschen Theorems, Math. Z. 24 (1925), 401-421. https://doi.org/10.1007/BF01216792
    • F. Jordan, Consonant spaces of countable type and the Menger property, Topology Appl. 341 (2024), 108755. https://doi.org/10.1016/j.topol.2023.108755
    • W. Just, A. W. Miller, Marion Scheepers, and P. J. Szeptycki, The combinatorics of open covers. II, Topology Appl. 73, no. 3 (1996), 241-266....
    • Lj. D. R. Kočinac, Selected results on selection principles, Proceedings of the Third Seminar on Geometry and Topology (Tabriz, Iran), July...
    • R. Laver, On the consistency of Borel's conjecture, Acta Math. 137 (1977), 151-169. https://doi.org/10.1007/BF02392416
    • R. A. McCoy, K-space function spaces, Int. J. Math. Math. Sci. 3 (1980), 701-711. https://doi.org/10.1155/S0161171280000506
    • R. A. McCoy, Function spaces which are k-spaces, Topology Proc. 5 (1981), 139-146.
    • R. M. Mezabarba, Pointless proofs of the Menger and Rothberger games, Topology Appl. 300 (2021), 107774. https://doi.org/10.1016/j.topol.2021.107774
    • E. Michael, Topologies on spaces of subsets, Trans. Amer. Math. Soc. 71 (1951), 152-182. https://doi.org/10.1090/S0002-9947-1951-0042109-4
    • A. W. Miller, Special Subsets of the Real Line, Handbook of Set-Theoretic Topology (Kenneth Kunen and Jerry E. Vaughan, eds.), North-Holland,...
    • A. V. Osipov, Selection principles in function spaces with the compact-open topology, Filomat 32, no. 15 (2018), 5403-5413. https://doi.org/10.2298/FIL1815403O
    • J. Pawlikowski, Undetermined sets of point-open games, Fundam. Math. 144, no. 3 (1994), 279-285. https://doi.org/10.4064/fm-144-3-279-285
    • π-base community database of topological counterexamples, https://topology.pi-base.org/.
    • I. Reclaw, Every Lusin set is undetermined in the point-open game, Fundam. Math. 144, no.1 (1994), 43-54. https://doi.org/10.4064/fm-144-1-43-54
    • F. Rothberger, Eine Verschärfung der Eigenschaft C, Fundam. Math. 30 (1938), 50-55. https://doi.org/10.4064/fm-30-1-50-55
    • F. Rothberger, Sur les familles indenombrables de suites de nombres naturels et les problèmes concernant la propriété C, Proc. Camb. Philos....
    • M. Sakai, Property C'' and function spaces, Proc. Amer. Math. Soc. 104, no. 3 (1988), 917-919. https://doi.org/10.1090/S0002-9939-1988-0964873-0
    • M. Scheepers, Combinatorics of open covers. I: Ramsey theory, Topology Appl. 69, no. 1 (1996), 31-62. https://doi.org/10.1016/0166-8641(95)00067-4
    • M. Scheepers, Combinatorics of open covers. III: Games, C_ p(X), Fundam. Math. 152, no. 3 (1997), 231-254.
    • M. Scheepers, A sequential property of C_p(X) and a covering property of Hurewicz, Proc. Amer. Math. Soc. 125, no. 9 (1997), 2789-2795. https://doi.org/10.1090/S0002-9939-97-03897-5
    • M. Scheepers, Lusin sets, Proc. Amer. Math. Soc. 127, no. 1 (1999), 251-257. https://doi.org/10.1090/S0002-9939-99-04512-8
    • M. Scheepers, Selection principles in topology: New directions, Filomat 15 (2001), 111-126.
    • M. Scheepers, Selection principles and covering properties in topology, Note Mat. 22, no. 2 (2003), 3-41.
    • W. M. Schmidt, On badly approximable numbers and certain games, Trans. Amer. Math. Soc. 123 (1966), 178-199. https://doi.org/10.1090/S0002-9947-1966-0195595-4
    • L. A. Steen and J. A. Seebach, Counterexamples in topology, Dover books on mathematics, Dover Publications, 1995.
    • P. Szewczak and B. Tsaban, Conceptual proofs of the Menger and Rothberger games, Topology Appl. 272 (2020), 107048. https://doi.org/10.1016/j.topol.2019.107048
    • P. Szewczak, B. Tsaban, and L. Zdomskyy, Finite powers and products of Menger sets, Fundam. Math. 253, no. 3 (2021), 257-275. https://doi.org/10.4064/fm896-4-2020
    • R. Telgársky, Topological games: on the 50th anniversary of the Banach-Mazur game, Rocky Mt. J. Math. 17 (1987), 227-276. https://doi.org/10.1216/RMJ-1987-17-2-227
    • L. Zdomskyy, Products of Menger spaces in the Miller model, Adv. Math. 335 (2018), 170-179. https://doi.org/10.1016/j.aim.2018.06.016
    • Ly. Zdomskyy, Selection principles in the Laver, Miller, and Sacks models, Centenary of the Borel conjecture, Providence, RI: American Mathematical...
Fundación Dialnet

Mi Documat

Opciones de artículo

Opciones de compartir

Opciones de entorno