×

Some covering properties in topological and uniform spaces. (English) Zbl 1351.54012

Proc. Steklov Inst. Math. 252, 122-137 (2006) and Tr. Mat. Inst. Steklova 252, 134-149 (2006).
Summary: Recent progress in selection principles theory is discussed and is illustrated mainly by the Hurewicz covering property and its strong version, the Gerlits-Nagy property \(\mathsf{GN}(\ast)\). Some results that have not been published elsewhere are given with proofs.

MSC:

54D20 Noncompact covering properties (paracompact, Lindelöf, etc.)
54E15 Uniform structures and generalizations
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] A. V. Arkhangel’skiĭ, ”The Frequency Spectrum of a Topological Space and the Classification of Spaces,” Dokl. Akad. Nauk SSSR 206, 265–268 (1972) [Sov. Math., Dokl. 13, 1185–1189 (1972)].
[2] A. V. Arkhangel’skiĭ, Topological Function Spaces (Moscow State Univ., Moscow, 1989; Kluwer, Dordrecht, 1992).
[3] A. V. Arhangel’skii, ”Projective {\(\sigma\)}-Compactness, {\(\omega\)} 1-Caliber, and Cp-Spaces,” Topol. Appl. 104, 13–26 (2000). · Zbl 0944.54012 · doi:10.1016/S0166-8641(99)00011-5
[4] L. Babinkostova, ”Metrizable Groups and Strict o-Boundedness,” Mat. Vesnik (in press), available at http://math.boisestate.edu/:_liljanab/Finalobounded.pdf · Zbl 1140.54016
[5] L. Babinkostova, ”Selective Screenability Game and Covering Dimension,” Topol. Proc. 29, 13–17 (2005). · Zbl 1148.54327
[6] L. Babinkostova, Lj. D. R. Kočinac, and M. Scheepers, ”Combinatorics of Open Covers (VIII),” Topol. Appl. 140, 15–32 (2004). · Zbl 1051.54019 · doi:10.1016/j.topol.2003.08.019
[7] L. Babinkostova, Lj. D. R. Kočinac, and M. Scheepers, ”Notes on Selection Principles in Topology (I): Paracompactness,” J. Korean Math. Soc. 42, 709–721 (2005). · Zbl 1077.54012 · doi:10.4134/JKMS.2005.42.4.709
[8] L. Babinkostova, Lj. D. R. Kočinac, and M. Scheepers, ”Combinatorics of Open Covers (XI): Menger-and Rothberger-Bounded Groups,” Topol. Appl. (in press). · Zbl 1114.54023
[9] L. Babinkostova and M. Scheepers, ”Combinatorics of Open Covers (IX): Basic Properties,” Note Mat. 22(2), 167–178 (2003). · Zbl 1115.54009
[10] L. Babinkostova and M. Scheepers, ”Combinatorics of Open Covers (X): Measure-like Properties of Metric Spaces,” Preprint, http://math.boisestate.edu/:_liljanab/Comb10.pdf · Zbl 1115.54009
[11] T. Banakh, ”Locally Minimal Topological Groups and Their Embeddings into Products of o-Bounded Groups,” Comment. Math. Univ. Carol. 41, 811–815 (2002). · Zbl 1049.54034
[12] T. Banakh, ”On Index of Total Boundedness of (Strictly) o-Bounded Groups,” Topol. Appl. 120, 427–439 (2002). · Zbl 1010.22004 · doi:10.1016/S0166-8641(01)00084-0
[13] T. Bartoszynski and S. Shelah, ”Continuous Images of Sets of Reals,” Topol. Appl. 116, 243–253 (2001). · Zbl 0992.03061 · doi:10.1016/S0166-8641(00)00079-1
[14] T. Bartoszynski and B. Tsaban, ”Hereditary Topological Diagonalizations and the Menger-Hurewicz Conjectures,” Proc. Am. Math. Soc. 134, 605–615 (2006). · Zbl 1137.54018 · doi:10.1090/S0002-9939-05-07997-9
[15] J. E. Baumgartner and A. D. Taylor, ”Partition Theorems and Ultrafilters,” Trans. Am. Math. Soc. 241, 283–309 (1978). · Zbl 0386.03024 · doi:10.1090/S0002-9947-1978-0491193-1
[16] M. Bonanzinga, F. Cammaroto, and Lj. D. R. Kočinac, ”Star-Hurewicz and Related Properties,” Appl. Gen. Topol. 5, 79–89 (2004). · Zbl 1080.54009 · doi:10.4995/agt.2004.1996
[17] E. Borel, ”Sur la classification des ensembles de mesure nulle,” Bull. Soc. Math. France 47, 97–125 (1919). · JFM 47.0181.02 · doi:10.24033/bsmf.996
[18] L. Bukovsky and J. Haleš, ”Hurewicz Properties,” Topol. Appl. 132, 71–79 (2003). · Zbl 1056.54024 · doi:10.1016/S0166-8641(02)00364-4
[19] A. Caserta, G. Di Maio, Lj. D. R. Kočinac, and E. Meccariello, ”Applications of k-Covers. II,” Topol. Appl. (in press). · Zbl 1117.54034
[20] G. Di Maio, Lj. D. R. Kočinac, and E. Meccariello, ”Applications of k-Covers,” Acta Math. Sinica, Engl. Ser. 22 (2006) (in press). · Zbl 1117.54034
[21] G. Di Maio, Lj. D. R. Kočinac, and E. Meccariello, ”Selection Principles and Hyperspace Topologies,” Topol. Appl. 153, 912–923 (2005). · Zbl 1087.54007 · doi:10.1016/j.topol.2005.01.020
[22] R. Engelking, General Topology (PWN, Warszawa, 1977; Mir, Moscow, 1986).
[23] J. Fell, ”A Hausdorff Topology for the Closed Subsets of a Locally Compact Non-Hausdorff Space,” Proc. Am. Math. Soc. 13, 472–476 (1962). · Zbl 0106.15801 · doi:10.1090/S0002-9939-1962-0139135-6
[24] J. Gerlits and Zs. Nagy, ”Some Properties of C(X). I,” Topol. Appl. 14, 151–161 (1982). · Zbl 0503.54020 · doi:10.1016/0166-8641(82)90065-7
[25] C. Guido and Lj. D. R. Kočinac, ”Relative Covering Properties,” Quest. Answers Gen. Topol. 19(1), 107–114 (2001). · Zbl 0991.54023
[26] C. Hernández, ”Topological Groups Close to Be {\(\sigma\)}-Compact,” Topol. Appl. 102, 101–111 (2000). · Zbl 0942.22001 · doi:10.1016/S0166-8641(98)00129-1
[27] C. Hernández, D. Robbie, and M. Tkachenko, ”Some Properties of o-Bounded and Strictly o-Bounded Groups,” Appl. Gen. Topol. 1, 29–43 (2000). · Zbl 0971.54026 · doi:10.4995/agt.2000.3022
[28] W. Hurewicz, ”Über die Verallgemeinerung des Borelschen Theorems,” Math. Z. 24, 401–425 (1925). · JFM 51.0454.02 · doi:10.1007/BF01216792
[29] W. Hurewicz, ”Über Folgen stetiger Funktionen,” Fundam. Math. 9, 193–204 (1927). · JFM 53.0562.03 · doi:10.4064/fm-9-1-193-210
[30] W. Just, A. W. Miller, M. Scheepers, and P. J. Szeptycki, ”The Combinatorics of Open Covers. II,” Topol. Appl. 73, 241–266 (1996). · Zbl 0870.03021 · doi:10.1016/S0166-8641(96)00075-2
[31] Lj. Kočinac, ”Star-Menger and Related Spaces,” Publ. Math. Debrecen 55, 421–431 (1999). · Zbl 0932.54022
[32] Lj. D. R. Kočinac, ”Closure Properties of Function Spaces,” Appl. Gen. Topol. 4(2), 255–261 (2003). · Zbl 1055.54007 · doi:10.4995/agt.2003.2030
[33] Lj. D. R. Kočinac, ”Selection Principles in Uniform Spaces,” Note Mat. 22(2), 127–139 (2003).
[34] Lj. D. R. Kočinac, ”Generalized Ramsey Theory and Topological Properties: A Survey,” Rend. Semin. Mat. Messina, II Ser. 25(9), 119–132 (2003) (Proc. Int. Symp. on Graphs, Designs and Applications, Messina, Sept. 30–Oct. 4, 2003)).
[35] Lj. D. R. Kočinac, ”Selected Results on Selection Principles,” in Proc. 3rd Semin. on Geometry and Topology, Tabriz, Iran, July 15–17, 2004, Ed. by Sh. Rezapour (Azarb. Univ. Tarbiat Moallem, Tabriz, 2004), pp. 71–104.
[36] Lj. D. R. Kočinac, ”The Reznichenko Property and the Pytkeev Property in Hyperspaces,” Acta Math. Hung. 107, 225–233 (2005). · Zbl 1082.54007 · doi:10.1007/s10474-005-0192-0
[37] Lj. D. R. Kočinac, ”{\(\gamma\)}-Sets, {\(\gamma\)} {\(\kappa\)}-Sets and Hyperspaces,” Math. Balkanica 19, 109–118 (2005).
[38] Lj. D. R. Kočinac, ”Selection Principles and Continuous Images,” Cubo Math. J. 8 (2006).
[39] Lj. D. R. Kočinac, ”Selection Principles Related to {\(\alpha\)}i Properties,” submitted to Taiwanese J. Math.
[40] Lj. D. R. Kočinac and L. Babinkostova, ”Function Spaces and Some Relative Covering Properties,” Far East J. Math. Sci., Spec. Vol., Part 2, 247–255 (2000). · Zbl 0991.54022
[41] Lj. D. R. Kočinac, C. Guido, and L. Babinkostova, ”On Relative {\(\gamma\)}-Sets,” East-West J. Math. 2(2), 195–199 (2000). · Zbl 0966.54009
[42] Lj. Kočinac and M. Scheppers, ”Function Spaces and Strong Measure Zero Sets,” Acta Math. Hung. 82, 341–351 (1999). · Zbl 0937.46022 · doi:10.1023/A:1006600627664
[43] Lj. D. Kočinac and M. Scheepers, ”Function Spaces and a Property of Reznichenko,” Topol. Appl. 123(1), 135–143 (2002). · Zbl 1011.54017 · doi:10.1016/S0166-8641(01)00177-8
[44] Lj. D. R. Kočinac and M. Scheepers, ”Combinatorics of Open Covers (VII): Groupability,” Fundam. Math. 179(2), 131–155 (2003). · Zbl 1115.91013 · doi:10.4064/fm179-2-2
[45] K. Menger, ”Einige Überdeckungssätze der Punktmengenlehre,” Sitzungsber. Wien. Akad., Abt. 2a: Math., Astron., Phys., Meteorol., Mech. 133, 421–444 (1924). · JFM 50.0129.01
[46] A. W. Miller, ”The Cardinal Characteristic for Relative {\(\gamma\)}-Sets,” Topol. Appl. (in press), available at http://www.math.wisc.edu/:_miller/res/relgamma.pdf · Zbl 1169.54004
[47] A. W. Miller and D. Fremlin, ”On Some Properties of Hurewicz, Menger and Rothberger,” Fundam. Math. 129, 17–33 (1988). · Zbl 0665.54026 · doi:10.4064/fm-129-1-17-33
[48] F. P. Ramsey, ”On a Problem of Formal Logic,” Proc. London Math. Soc. 30, 264–286 (1930). · JFM 55.0032.04 · doi:10.1112/plms/s2-30.1.264
[49] F. Rothberger, ”Eine Verschärfung der Eigenschaft C,” Fundam. Math. 30, 50–55 (1938). · JFM 64.0622.01
[50] F. Rothberger, ”Sur les familles indénombrables des suites de nombres naturels et les problémes concernant la propriété C,” Proc. Cambridge Philos. Soc. 37, 109–126 (1941). · JFM 67.0990.01 · doi:10.1017/S0305004100021617
[51] M. Sakai, ”Property C” and Function Spaces,” Proc. Am. Math. Soc. 104, 917–919 (1988). · Zbl 0691.54007
[52] M. Scheepers, ”Combinatorics of Open Covers. I: Ramsey Theory,” Topol. Appl. 69, 31–62 (1996). · Zbl 0848.54018 · doi:10.1016/0166-8641(95)00067-4
[53] M. Scheepers, ”Selection Principles and Covering Properties in Topology,” Note Mat. 22(2), 3–41 (2003). · Zbl 1195.37029
[54] W. Sierpiński, ”Sur un problème de K. Menger,” Fundam. Math. 8, 223–224 (1926). · JFM 52.0198.01 · doi:10.4064/fm-8-1-223-224
[55] W. Sierpiński, ”Sur un ensemble nondénombrable, donc toute image continue est de mesure nulle,” Fundam. Math. 11, 301–304 (1928). · JFM 54.0097.03
[56] SPM Bulletin 11 (2004), math.GN/0412305.
[57] M. Talagrand, ”A New Countably Determined Banach Space,” Isr. J. Math. 47, 75–80 (1984). · Zbl 0537.46019 · doi:10.1007/BF02760563
[58] B. Tsaban, ”A Diagonalization Property between Hurewicz and Menger,” Real Anal. Exch. 27, 1–7 (2001/2002). · Zbl 1044.26001
[59] B. Tsaban, ”Selection Principles in Mathematics: A Milestone of Open Problems,” Note Mat. 22(2), 179–208 (2003). · Zbl 1223.37059
[60] B. Tsaban, ”o-Bounded Groups and Other Topological Groups with Strong Combinatorial Properties,” Proc. Am. Math. Soc. 134, 881–891 (2006). · Zbl 1090.54034 · doi:10.1090/S0002-9939-05-08034-2
[61] B. Tsaban, ”Strong {\(\gamma\)}-Sets and Other Singular Spaces,” Topol. Appl. 153, 620–639 (2005). · Zbl 1094.54010 · doi:10.1016/j.topol.2005.01.033
[62] B. Tsaban, ”Some New Directions in Infinite-Combinatorial Topology,” Topics in Set Theory and Its Applications, Ed. by J. Bagaria and S. Todorčević (Birkhäuser, Basel, in press); math.GN/0409069. · Zbl 1113.54002
[63] L. Zdomsky, ”A Semifilter Approach to Selection Principles,” Comment. Math. Univ. Carol. 46, 525–540 (2005). · Zbl 1121.03060
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.