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The Combinatorics of Open Covers

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Recent Progress in General Topology III

Abstract

The combinatorics of open covers is a study of Cantor’s diagonal argument in various contexts. The field has its roots in a few basic selection principles that arose from the study of problems in analysis, dimension theory, topology and set theory. The reader will also find that some familiar works are appearing in new clothes in our survey. This is particularly the case in connection with such problems as determining the structure of compact scattered spaces and a number of classical problems in topology. We hope that the new perspective in which some of these classical enterprises are presented will lead to further progress. In this article we also attempt to give the reader an overview of the problems and techniques that are currently fueling much of the rapidly increasing current activity in the combinatorics of open covers.

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Notes

  1. 1.

    Strictly speaking, the definition we give here equivalent to but not he original one given by Alexandroff. This will be clarified when Alexandroff’s original definition is given below in Sect. 6.

  2. 2.

    The operation \(\oplus \) on \(^{\kappa }2\) is coordinate-wise addition modulo 2.

  3. 3.

    That is, the topology induced by the metric \(p\) on \([0,\, 1]^{R}\), where \(p(f, g) := \sup \{\vert f(y) -g(y)\vert : y\in R\}\).

  4. 4.

    See [174], Proposition 8.

  5. 5.

    We use the following abbreviations: CC = Corson compact, EC = Eberlein compact, SEC = strong Eberlein compact, RNC = Radon-Nikodým compact.

  6. 6.

    We define the notion of a Hurewicz space later. These are spaces with a certain covering property.

  7. 7.

    This covering property is investigated in [186].

  8. 8.

    This covering property is a cozero set version of \((\alpha _1)\) in Kočinac [108].

  9. 9.

    This was first given for strongly zero-dimensional perfectly normal spaces \(X\) in the first version of Tsaban and Zdomskyy [189].

  10. 10.

    “metrizable”can be replaced by “perfectly normal” [91].

  11. 11.

    “R” for Rothberger. Rothberger did not consider \(\mathsf{S}_1(\mathfrak {D},\mathfrak {D})\). It has become customary to use the “R” in connection with selection principles of the form \(\mathsf{S}_1(\fancyscript{A},\fancyscript{A})\) since Rothberger introduced the first prototype of such a selection principle.

  12. 12.

    This was first proved for separable metrizable spaces in [161].

  13. 13.

    We are taking the liberty of giving an equivalent reformulation of Menger’s original conjecture. Menger did not mention \(\mathsf{S}_{fin}({\fancyscript{O}},{\fancyscript{O}})\) at all.

  14. 14.

    Note that this example is not a subspace of the real line.

  15. 15.

    “M” here refers to Menger. It has become common to call selection principles of the form \(\mathsf{S}_{fin}({\fancyscript{A}},{\fancyscript{B}})\) an “M-property”, with reference to the Menger property \(\mathsf{S}_{fin}({\fancyscript{O}},{\fancyscript{O}})\) which was the first prototype of this type of selection principle. Note that Hurewicz actually introduced this prototype, and poved that in metrizable spaces it is equivalent to a basis property introduced by Menger.

  16. 16.

    This was first proved for separable metrizable spaces in [161].

  17. 17.

    Note that the definition for weakly infinite dimensionality given here is the original definition, while the definition we gave earlier in Sect. 3 is an equivalent form of the original definition.

  18. 18.

    \({\mathbb K}\) is the well-known example of R. Pol [134] which answered Alexandroff’s problem.

References

  1. T. Banakh and Lj. Zdomskyy, Selection principles and infinite games on multicovered spaces, in [7], 1–52.

    Google Scholar 

  2. T. Banakh and Lj. Zdomskyy, The coherence of semifilters: a Survey, in [7], 53–106.

    Google Scholar 

  3. J.E. Baumgartner and F.D. Tall, Reflecting Lindelöfness, Topology and its Applications 122 (2002), 35–49.

    Google Scholar 

  4. L. Bukovský, Convergence of Real Functions and Covering Properties, in [7], 107–132.

    Google Scholar 

  5. Lj.D.R. Kočinac, Generalized Ramsey Theory and topological properties: A survey, (Proceedings of the International symposium on graphs, designs and applications, Messina, Sept 30–Oct. 4 2003) Rendiconti del Seminario Matematico di Messina 25:II (2003), 119–132.

    Google Scholar 

  6. Lj.D.R. Kočinac, Selected results on selection principles (in Sh. Rezapour, ed.) Proceedings of the third seminar on geometry and topology (July 15–17, 2004, Tabriz, Iran), 71–104.

    Google Scholar 

  7. Lj.D.R. Kočinac, Selection principles and covering properties in Topology (Editor), Quaderni di Mathematica 18, 2006.

    Google Scholar 

  8. M. Sakai, Special subsets of reals characterizing local properties of function spaces, in [7], pp. 195–225.

    Google Scholar 

  9. M. Scheepers, Selection principles in Topology: New directions, Filomat (Nis) 17:2 (2003), 111–126.

    Google Scholar 

  10. M. Scheepers, Selection principles and covering properties in topology, Note di Matematica 22:2 (2003/2004), 3–41.

    Google Scholar 

  11. M. Scheepers, Topological games and Ramsey theory, in [129], 61–90.

    Google Scholar 

  12. M. Scheepers, Gerlits and function spaces, Studia Scientiarum Mathematicarum Hungarica 47:4 (2010), 529–557.

    Google Scholar 

  13. B. Tsaban, Selection principles in Mathematics: A milestone of open problems, Note di Matematica 22:2 (2003/2004), 179–208.

    Google Scholar 

  14. B. Tsaban, Some new directions in infinite combinatorial topology, in Set Theory (J. Bagaria and S. Todorčevic eds) Trends in Mathematics, Birkhaüser (2006), 225–255.

    Google Scholar 

  15. B. Tsaban, Selection principles and special sets of reals, in [129], 91–108.

    Google Scholar 

  16. B. Tsaban, Menger’s and Hurewicz’s problems: Solutions from “The Book” and refinements, Contemporary Mathematics 533 (2011), 211–226.

    Google Scholar 

  17. D.F. Addis and J.H. Gresham, A class of infinite-dimensional spaces Part I Dimension theory and Alexandroff’s problem, Fundamenta Mathematicae 104 (1978), 195–205.

    Google Scholar 

  18. O.T. Alas, L.F. Aurichi, L.R. Junqueira and F.D. Tall, Non-productively Lindelöf spaces and small cardinals, Houston Journal of Mathematics 37 (2011), 1373–1381.

    Google Scholar 

  19. P.S. Aleksandrof and B.A. Pasynkov, Introduction to dimension theory (in Russian), “Nauka” Moscow, 1973.

    Google Scholar 

  20. K. Alster, Some remarks on Eberlein compacts, Fundamenta Mathematicae 104 (1979), 43–46.

    Google Scholar 

  21. K. Alster, On a question of Archangelskij concerning Lindelöf spaces with countable pseudocharacter, Proceedings of the American Mathematical Society 95:2 (1985), 320–322.

    Google Scholar 

  22. A.V. Arhangelskii, On the cardinality of bicompacta satisfying the first axiom of countability, Doklady Acad. Nauk SSSR 187 (1969), 967–970 (in Russian).

    Google Scholar 

  23. A.V. Arhangel’skii, The frequency spectrum of a topological space and the classification of spaces, Soviet Mathematics. Doklady 13(1972), 1185–1189.

    Google Scholar 

  24. A.V. Arhangel’skii, On some topological spaces that arise in functional analysis, Russian Mathematical Surveys 31(1976), 14–30.

    Google Scholar 

  25. A.V. Arhangel’skii, Hurewicz spaces, analytic sets and fan tightness of function spaces, Soviet Mathematics. Doklady 33(1986), 396–399.

    Google Scholar 

  26. L. Aurichi, D-spaces, topological games and selection principles, Topology Proceedings 36(2010), 107–122.

    Google Scholar 

  27. L. Babinkostova, Selective versions of screenability, Filomat 17:2 (2003), 127–134.

    Google Scholar 

  28. L. Babinkostova, Selective screenability game and covering dimension, Topology Proceedings 29:1 (2005), 1–5.

    Google Scholar 

  29. L. Babinkostova, Strictly \(o\) -bounded metrizable groups, Mathematički Vesnik 58:3–4 (2006), 131–138.

    Google Scholar 

  30. L. Babinkostova, When does the Haver property imply selective screenability?,  Topology and its Applications 154:9 (2007), 1971–1979.

    Google Scholar 

  31. L. Babinkostova, Selective screenability in topological groups, Topology and its Applications 156:1 (2008), 2–9.

    Google Scholar 

  32. L. Babinkostova, Selective screenability and the Hurewicz property, Topology Proceedings 32 (2008), 245–252.

    Google Scholar 

  33. L. Babinkostova, On some questions about selective separability, Mathematical Logic Quarterly 55 (2009), 539–541.

    Google Scholar 

  34. L. Babinkostova, Topological games and covering dimension, Topology Proceedings 38 (2011), 99–120.

    Google Scholar 

  35. L. Babinkostova, Topological groups and covering dimension, Topology and its Applications 158 (2011), 1460–1470.

    Google Scholar 

  36. L. Babinkostova, Lj.D.R. Kočinac and M. Scheepers, Combinatorics of open covers (XI): Menger- and Rothberger-bounded groups, Topology and its Applications 154 (2007), 1269–1280.

    Google Scholar 

  37. L. Babinkostova, B.A. Pansera and M. Scheepers, Weak covering properties and infinite games, Topology and its Applications, to appear.

    Google Scholar 

  38. L. Babinkostova and M. Scheepers, Products and selection principles, Topology Proceedings 31 (2007), 431–443.

    Google Scholar 

  39. S. Baldwin, Possible point-open types of subsets of the reals, Topology and its Applications 38 (1991), 219–223.

    Google Scholar 

  40. D. Barman, A. Dow, Selective separability and SS\(^+\), Topology Proceedings 37 (2011), 181–204.

    Google Scholar 

  41. D. Barman, A. Dow, Proper forcing axiom and selective separability, Topology and its Applications 159 (2012), 806–813.

    Google Scholar 

  42. A. Bella, M. Bonanzinga and M.V. Matveev, Variations of selective separability, Topology and its Applications 156 (2009), 1241–1252.

    Google Scholar 

  43. A. Bella, M. Bonanzinga, M.V. Matveev and V.V. Tkachuk, Selective separability: General facts and behavior in countable spaces,  Topology Proceedings 32 (2008), 15–30.

    Google Scholar 

  44. A. Bella, M. Matveev and S. Spadaro, Variations of selective separability II: Discrete sets and the influence of convergence and maximality, Topology and its Applications 159:1 (2012), 253–271.

    Google Scholar 

  45. Y. Benyamini, M.E. Rudin and M. Wage,Continuous images of weakly compact subsets of Banach spaces, Pacific Journal of Mathematics 70 (1977), 309–324.

    Google Scholar 

  46. A. Berner and I. Juhasz, Point-picking games and HFD’s, in: Models and Sets, Proceedings of the Logic Colloquium 1983, Springer-Verlag Lecture Notes in Mathematics 1103 (1984), 53–66.

    Google Scholar 

  47. R.H. Bing, Metrization of topological spaces, Canadian Journal of Mathematics 3 (1951), 175–186.

    Google Scholar 

  48. M. Bonanzinga and M. Matveev, On spaces where point-cofinite covers can be diagonalized, preprint.

    Google Scholar 

  49. E. Borel, Sur la classification des ensembles de mesure nulle, Bulletin de la Societe Mathematique de France 47 (1919), 97–125.

    Google Scholar 

  50. P. Borst, Classification of weakly infinite dimensional spaces Part I: A transfinite extension of covering dimension, Fundamenta Mathematicae 130 (1988), 1–25.

    Google Scholar 

  51. P. Borst, Some remarks concerning C-spaces, Topology and its Applications 154 (2007), 665–674.

    Google Scholar 

  52. C. Brech and P. Koszmider, Thin-very tall compact scattered spaces which are hereditarily separable, Transactions of the American Mathematical Society 363:1 (2011), 501–519.

    Google Scholar 

  53. Z. Bukovská, Thin sets in trigonometrical series and quasinormal convergence, Mathematica Slovaca 40(1990), 53–62.

    Google Scholar 

  54. Z. Bukovská, Quasinormal convergence, Mathematica Slovaca 41(1991), 137–146.

    Google Scholar 

  55. L. Bukovský, Hurewicz properties, not distinguishing convergence properties and sequence selection properties, Acta Universitatis Carolinae. Mathematica et Physica 44(2003), 45–56.

    Google Scholar 

  56. L. Bukovský,On wQN \(_*\) and wQN \(^*\) spaces, Topology and its Applications 156 (2008), 24–27.

    Google Scholar 

  57. L. Bukovský and J. Haleš, On Hurewicz properties, Topology and its Applications 132(2003), 71–79.

    Google Scholar 

  58. L. Bukovský and J. Haleš, QN-space, wQN-space and covering properties, Topology and its Applications 154 (2007), 848–858.

    Google Scholar 

  59. L. Bukovský, I. Recław and M. Repický, Spaces not distinguishing pointwise and quasinormal convergence of real functions, Topology and its Applications 41 (1991), 25–40.

    Google Scholar 

  60. L. Bukovský, I. Recław and M. Repický, Spaces not distinguishing convergences of real-valued functions, Topology and its Applications 112 (2001), 13–40.

    Google Scholar 

  61. L. Bukovský, J. Šupina, Sequence selection principles for quasi-normal convergence, Topology and its Applications 159 (2012), 283–239.

    Google Scholar 

  62. T.J. Carlson, Strong measure zero and strongly meager sets, Proceedings of the American Mathematical Society 118:2 (1993), 577–586.

    Google Scholar 

  63. J. Chaber and R. Pol, A remark on Fremlin-Miller Theorem concerning the Menger property and Michael concentrated sets, Oct. 2002, unpublished manuscript.

    Google Scholar 

  64. Á. Császár and M. Laczkovich, Discrete and equal convergence, Studia Scientiarum Mathematicarum Hungarica 10 (1975), 463–472.

    Google Scholar 

  65. Á. Császár and M. Laczkovich, Some remarks on discrete Baire classes, Acta Mathematica Academiae Scientiarum Hungaricae 33 (1979), 51–70.

    Google Scholar 

  66. P. Daniels, Pixley-Roy spaces over subsets of the reals, Topology and its Applications 29 (1988), 93–106.

    Google Scholar 

  67. P. Daniels and G. Gruenhage, The point-open type of subsets of the reals, Topology and its Applications 37 (1990), 53–64.

    Google Scholar 

  68. G. Di Maio, Lj.D. Kočinac and E. Meccariello, Selection principles and hyperspace topologies, Topology and its Applications, 153 (2005), 912–923.

    Google Scholar 

  69. G. Di Maio and Lj.D. Kočinac, Some covering properties of hyperspaces, Topology and its Applications155 (2008), 1959–1969.

    Google Scholar 

  70. E.K. van Douwen, The Pixley-Roy topology on spaces of subsets, Set Theoretic Topology, Academic Press, 1977, pp. 111–134.

    Google Scholar 

  71. A. Dow, Two classes of Fréchet-Urysohn spaces, Proceedings of the American Mathematical Society 108 (1990), 241–247.

    Google Scholar 

  72. S. Eilenberg and E. Otto, Quelques propriétés caracteristiques de la dimension, Fundamenta Mathematicae 31 (1938), 149–153.

    Google Scholar 

  73. E. Ellentuck, A new proof that analytic sets are Ramsey, Journal of Symbolic Logic 39 (1) (1974), 163–165.

    Google Scholar 

  74. R. Engelking, Theory of dimensions finite and infinite, Heldermann Verlag, 1995.

    Google Scholar 

  75. V.V. Fedorchuk, Weakly infinite-dimensional spaces, Russian Mathematical Surveys 42 (2) (2007), 1–52.

    Google Scholar 

  76. V.V. Fedorchuk, Questions on weakly infinite-dimensional spaces, in: [129], pp. 637–645.

    Google Scholar 

  77. D.H. Fremlin, Sequential convergence in \({ C}_p(X)\), Commentationes Mathematicae Universitatis Carolinae 35 (1994), 371–382.

    Google Scholar 

  78. D.H. Fremlin, SSP and wQN, manuscript.

    Google Scholar 

  79. D.H. Fremlin and A.W. Miller, On some properties of Hurewicz, Menger, and Rothberger, Fundamenta Mathematicae 129(1988), 17–33.

    Google Scholar 

  80. F. Galvin, Indeterminacy of the point-open game, Bulletin de L’Academie Polonaise des Sciences, Serie des sciences, math., astr, phys. 26:5 (1978), 445–449.

    Google Scholar 

  81. F. Galvin and A.W. Miller, \(\gamma \) -sets and other singular sets of real numbers, Topology and its Applications 17 (1984), 145–155.

    Google Scholar 

  82. F. Galvin, J. Mycielski and R. Solovay, Notices of the American Mathematical Society  26 (1979), Abstract A-280.

    Google Scholar 

  83. F. Galvin and K. Prikry, Borel sets and Ramsey’s Theorem, Journal of Symbolic Logic 38 (2) (1973), 193–198.

    Google Scholar 

  84. F. Galvin and M. Scheepers, Borel’s Conjecture in topological groups, preprint.

    Google Scholar 

  85. J. Gerlits, Some properties of \(C(X)\) , II,  Topology and its Applications 15(1983), 255–262.

    Google Scholar 

  86. J. Gerlits and Zs. Nagy, Some properties of C(X), I, Topology and its Applications 14 (1982), 151–161.

    Google Scholar 

  87. I. Gorelic, The Baire category and forcing large Lindelöf spaces with points \({ G}_{\delta }\), Proceedings of the American Mathematical Society  118:2 (1993), 603–607.

    Google Scholar 

  88. G. Gruenhage and M. Sakai, Selective separability and its variations, Topology and its Applications 158 (2011), 1352–1359.

    Google Scholar 

  89. I.I. Guran, On topological groups close to being Lindelöf, Soviet Mathematics. Doklady 23 (1981), 173–175.

    Google Scholar 

  90. A. Hajnal and I. Juhász, Lindelöf spaces a la Shelah, Colloquium Mathematicum Societatis János Bolyai 23 (1978), 555–567.

    Google Scholar 

  91. J. Haleš, On Scheepers’ conjecture, Acta Universitatis Carolinae. Mathematica et Physica 46 (2005), 27–31.

    Google Scholar 

  92. Y. Hattori and K. Yamada, Closed pre-images of C-spaces, Mathematica Japonica 34:4 (1989), 555–561.

    Google Scholar 

  93. W.E. Haver, A covering property for metric spaces, Lecture Notes in Mathematics (Springer-Verlag) 375 (1974), 108–113.

    Google Scholar 

  94. W. Hurewicz, Über eine verallgemeinerung des Borelschen Theorems, Mathematische Zeitschrift 24 (1925), 401–421.

    Google Scholar 

  95. W. Hurewicz, Über Folgen stetiger Funktionen, Fundamenta Mathematicae 9 (1927), 193–204.

    Google Scholar 

  96. W. Hurewicz, Normalbereiche und Dimensionstheorie, Mathematische Annalen 96 (1927), 736–764.

    Google Scholar 

  97. W. Hurewicz, Ueber unendlich-dimensionale Punktmengen, Proc. Akad. Amsterdam 31 (1928), 916–922.

    Google Scholar 

  98. F. Jordan, There are no hereditary productive \(\gamma \) -spaces, Topology and its Applications 155 (2008), 1786–1791.

    Google Scholar 

  99. I. Juhász, HFD and HFC type spaces, with applications, Topology and its Applications 126 (2002), 217–262.

    Google Scholar 

  100. I. Juhász and S. Shelah, \(\pi (X)=\delta (X)\) for compact \(X\), Topology and its Applications 32 (1989), 289–294.

    Google Scholar 

  101. I. Juhász and W.A.R. Weiss, On thin-tall scattered spaces, Colloquium Mathematicum 40:1 (1978/79), 63–68.

    Google Scholar 

  102. L.R. Junqueira and F.D. Tall, More reflections on compactness, Fundamenta Mathematicae 176 (2003), 127–141.

    Google Scholar 

  103. W. Just, Two consistency results concerning thin-tall Boolean algebras,  Algebra Universalis 20 (1985), 135–142.

    Google Scholar 

  104. W. Just, A.W. Miller, M. Scheepers and P.J. Szeptycki, The combinatorics of open covers II, Topology and its Applications 73 (1996), 241–266.

    Google Scholar 

  105. M. Kada, Preserving the Lindelof property under forcing extensions, Topology Proceedings 38 (2011), 237–251.

    Google Scholar 

  106. D. Kocev, Almost Menger and related spaces, Matematički Vesnik 61(2009), 173–180.

    Google Scholar 

  107. Lj.D.R. Kočinac, Selection principles in uniform spaces, Note di Matematica 22:2 (2003/2004), 127–139.

    Google Scholar 

  108. Lj.D. Kočinac, Selection principles related to \(\alpha _i\) -properties, Taiwanese Journal of Mathematics 12 (2008), 561–571.

    Google Scholar 

  109. Lj.D. Kočinac and M. Scheepers, Function spaces and a property of Reznichenko, Topology and its Applications 123 (2002), 135–143.

    Google Scholar 

  110. Lj.D.R. Kočinac and M. Scheepers, Combinatorics of open covers (VII): Groupability, Fundamenta Mathematicae 179 (2003), 131–155.

    Google Scholar 

  111. P. Koszmider and F.D. Tall, A Lindelöf subspace of cardinality \(\aleph _1\), Proceedings of the American Mathematical Society 130:9 (2002), 2777–2787.

    Google Scholar 

  112. R. Laver, On the consistency of Borel’s conjecture, Acta Mathematicae 137 (1976), 151–169.

    Google Scholar 

  113. V.I. Malykhin, G. Tironi, Weakly Fréchet-Urysohn and Pytkeev spaces, Topology and its Applications 104 (2000), 181–190.

    Google Scholar 

  114. J. C. Martinez, A consistency result on thin-very tall Boolean algebras, Israel Journal of Mathematics, 123 (2001) 273–284.

    Google Scholar 

  115. S. Mazurkiewicz and W. Sierpiński, Contribution à la topologie des ensembles dénombrables, Fundamenta Mathematicae 1 (1920), 2–27.

    Google Scholar 

  116. K. Menger, Einige Überdeckungssätze der Punktmengenlehre, Sitzungsberichte Abt. 2a, Mathematik, Astronomie, Physik, Meteorologie und Mechanik 133 (1924), 421–444.

    Google Scholar 

  117. E. A. Michael, Paracompactness and the Lindelöf property in fnite and countable Cartesian products, Compositio Mathematicae 23 (1971), 199–214.

    Google Scholar 

  118. A.W. Miller, The \(\gamma \) -Borel Conjecture, Archive for Mathematical Logic 44:4 (2005), 425–434.

    Google Scholar 

  119. A.W. Miller and B. Tsaban, Point-cofinite covers in the Laver model, Proceedings of the American Mathematical Sociey 138 (2010), 3313–3321.

    Google Scholar 

  120. I. Namioka, Radon-Nikodým compact spaces and fragmentability, Mathematika 34 (1987), 258–281.

    Google Scholar 

  121. A. Nowik, M. Scheepers and T. Weiss, The algebraic sum of sets of real numbers with strong measure zero sets, The Journal of Symbolic Logic 63 (1998), 301–324.

    Google Scholar 

  122. P.J. Nyikos, Subsets of \(^\omega \omega \) and the Fréchet-Urysohn and \(\alpha _i\) -properties, Topology and its Applications 48 (1992), 91–116.

    Google Scholar 

  123. P.J. Nyikos and S. Purisch, Monotone normality and paracompactness in scattered spaces, Annals of the New York Academy of Sciences 552(1989), 124–137.

    Google Scholar 

  124. H. Ohta, M. Sakai, Sequences of semicontinuous functions accompanying continuous functions, Topology and its Applications 156 (2009), 2683–2691.

    Google Scholar 

  125. T. Orenshtein and B. Tsaban, Linear sigma-additivity and some applications, Transactions of the American Mathematical Society 363 (2011), 3621–3637.

    Google Scholar 

  126. J. Orihuela, W. Schachermayer and M. Valdivia, Every Radon-Nikodym Corson compact space is Eberlein compact, Studia Mathematica 98:2 (1991), 157–174.

    Google Scholar 

  127. B.A. Pansera, Weaker forms of the Menger property, Quaestiones Mathematicae 35:2 (2012), 161–169.

    Google Scholar 

  128. J. Pawlikowski, Undetermined sets of point-open games, Fundamenta Mathematicae 144 (1994), 279–285.

    Google Scholar 

  129. E. Pearl (Editor), Open problems in Topology II, Elsevier, 2007.

    Google Scholar 

  130. C. Pixley and P. Roy, Uncompletable Moore spaces, Proceedings of the Auburn Topology Conference, (1969), 75–85.

    Google Scholar 

  131. E. Pol, A weakly infinite dimensional space whose product with the irrationals is strongly infinite dimensional, Proceedings of the American Mathematical Society

    Google Scholar 

  132. E. Pol, Spaces whose \(n\) -th power is weakly infinite dimensional, but whose \(n+1\) -th power is not, Proceedings of the American Mathematical Society 17:3 (1993), 871–876.

    Google Scholar 

  133. E. Pol and R. Pol, On metric spaces with the Menger property which are Haver spaces, Topology and its Applications 157 (2010), 1495–1505.

    Google Scholar 

  134. R. Pol, A weakly infinite-dimensional compactum which is not countable dimensional, Proceedings of the American Mathematical Society 82 (1981), 634–636.

    Google Scholar 

  135. R. Pol, A remark on A-weakly infinite-dimensional spaces, Topology and its Applications 13 (1982), 97–101.

    Google Scholar 

  136. R. Pol, On classification of weakly infinite dimensional compacta, Fundamenta Mathematicae 116 (1983), 169–188.

    Google Scholar 

  137. R. Pol, Weak infinite-dimensionality in Cartesian products with the Menger property, Topology and its Applications 61 (1995), 85–94.

    Google Scholar 

  138. T.C. Przymusiński, Normality and paracompactness of Pixley-Roy hyperspaces, Fundamenta Mathematicae 113(1981), 201–219.

    Google Scholar 

  139. E.G. Pytkeev, The tightness of spaces of continuous functions, Russian Mathematical Surveys 37 (1982), 176–177.

    Google Scholar 

  140. E.G. Pytkeev, Sequentiality of spaces of continuous functions, Uspehi Matematiceskih Nauk 37 (1982), 197–198 (in Russian).

    Google Scholar 

  141. F.P. Ramsey, On a problem of formal logic, Proceedings of the London Mathematical Society 30 (1930), 264–286.

    Google Scholar 

  142. I. Recław, Metric spaces not distinguishing pointwise and quasinormal convergence of real functions, Bulletin of the Polish Academy of Sciences 45 (1997), 287–289.

    Google Scholar 

  143. D. Repovš and L. Zdomskyy, On M-separability of countable spaces and function spaces, Topology and its Applications 157 (2010), 2538–2541.

    Google Scholar 

  144. D.M. Rohm, Products of infinite dimensional spaces, Proceedings of the American Mathematical Society 108:4 (1990), 1019–1023.

    Google Scholar 

  145. F. Rothberger, Eine Verschärfung der Eigenschaft C, Fundamenta Mathematicae 30 (1938), 50–55.

    Google Scholar 

  146. L.R. Rubin, R.M. Schori, J.J. Walsh, New dimension-theory techniques for constructing infinite-dimensional examples, General Topology and its Applications 10 (1979), 93–102.

    Google Scholar 

  147. M. Sakai, Property \({ C}^{\prime \prime }\) and function spaces, Proceedings of the American Mathematical Society 104:3 (1988), 917–919.

    Google Scholar 

  148. M. Sakai, The Pytkeev property and the Reznichenko property in function spaces, Note di Matematica 22 (2003/2004), 43–52.

    Google Scholar 

  149. M. Sakai, Two properties of \({ C}_p(X)\) weaker than the Fréchet Urysohn property, Topology and its Applications 153 (2006), 2795–2804.

    Google Scholar 

  150. M. Sakai, The sequence selection properties of \({ C}_p(X)\), Topology and its Applications 154 (2007), 552–560.

    Google Scholar 

  151. M. Sakai, Selection principles and upper semicontinuous functions, Colloquium Mathematicum 117 (2009), 251–256.

    Google Scholar 

  152. M. Sakai, The Ramsey property for \({ C}_p(X)\), Acta Mathematica Hungarica 128 (2010), 96–105.

    Google Scholar 

  153. M. Sakai, Selective separability of Pixley-Roy hyperspaces, Topology and its Applications 159 (2012), 1591–1598.

    Google Scholar 

  154. M. Scheepers, A topological space could have infinite successor point-open type, Topology and its Applications 61 (1995), 95–99.

    Google Scholar 

  155. M. Scheepers, The combinatorics of open covers (I): Ramsey theory, Topology and its Applications 69 (1996), 31–62.

    Google Scholar 

  156. M. Scheepers, Combinatorics of open covers (III): games, \({ C}_p(X)\), Fundamenta Mathematicae 152 (1997), 231–254.

    Google Scholar 

  157. M. Scheepers, A sequential property of \({ C}_p(X)\) and a covering property of Hurewicz, Proceedings of the American Mathematical Sociey 125 (1997), 2789–2795.

    Google Scholar 

  158. M. Scheepers, \({ C}_p(X)\) and Arhangel’skii’s \(\alpha _i\) -spaces, Topology and its Applications 89 (1998), 265–275.

    Google Scholar 

  159. M. Scheepers, Sequential convergence in \({ C}_p(X)\) and a covering property, East-West Journal of Mathematics 1 (1999), 207–214.

    Google Scholar 

  160. M. Scheepers, Open covers and partition relations, Proceedings of the American Mathematical Society 127:2 (1999), 577–581.

    Google Scholar 

  161. M. Scheepers, Combinatorics of open covers (VI): Selectors for sequences of dense sets, Quaestiones Mathematicae 22 (1999), 109–130.

    Google Scholar 

  162. M. Scheepers, Finite powers of strong measure zero sets, The Journal of Symbolic Logic 64:3 (1999), 1295–1306.

    Google Scholar 

  163. M. Scheepers, The length of some diagonalization games, Archive for Mathematical Logic 38 (1999), 103–122.

    Google Scholar 

  164. M. Scheepers, Combinatorics of open covers (V): Pixley-Roy spaces of sets of reals and \(\omega \) -covers, Topology and its Applications 102 (2000), 13–31.

    Google Scholar 

  165. M. Scheepers, Rothberger’s property in all finite powers, Topology and its Applications 156 (2008), 93–103.

    Google Scholar 

  166. M. Scheepers, Measurable cardinals and the cardinality of Lindelöf spaces, Topology and its Applications 157 (2010), 1651–1657.

    Google Scholar 

  167. M. Scheepers, Rothberger bounded groups and Ramsey Theory, Topology and its Applications 158 (2011), 1575–1583.

    Google Scholar 

  168. M. Scheepers, Remarks on countable tightness, ArXiv 1201.4909v1.

    Google Scholar 

  169. M. Scheepers and F.D. Tall, Lindelöf indestructibility, topological games and selection principles, Fundamenta Mathematicae  210 (2010), 1–46.

    Google Scholar 

  170. S. Shelah, On some problems in General Topology, Contemporary Mathematics  192 (1996), 91–101.

    Google Scholar 

  171. W. Sierpiński, Sur un problème de M. Menger, Fundamenta Mathematicae 8 (1926), 223–224.

    Google Scholar 

  172. W. Sierpiński, Sur un ensemble non-dénombrable, dont toute image continue est de mesure nulle, Fundamenta Mathematicae  11 (1928), 302–304.

    Google Scholar 

  173. W. Sierpiński, Sur le produit combinatoire de deux ensembles jouissant de la propriété C, Fundamenta Mathematicae  24 (1935), 48–50.

    Google Scholar 

  174. P. Simon, On continuous images of Eberlein compacts, Commentationes Mathematicae Universitatis Carolinae 17 (1976), 179–194.

    Google Scholar 

  175. P. Simon and B. Tsaban, On the Pytkeev property in spaces of continuous functions, Proceedings of the American Mathematical Society 136 (2008), 1125–1135.

    Google Scholar 

  176. D. Soukup and P.J. Szeptycki, A counterexample in the theory of D-spaces, (preprint).

    Google Scholar 

  177. E. Szpilrajn, Sur une hypothèse de M. Borel, Fundamenta Mathematicae 15:1 (1930), 126–127.

    Google Scholar 

  178. E. Szpilrajn, La dimension et la mesure, Fundamenta Mathematicae 28 (1937), 81–89.

    Google Scholar 

  179. F.D. Tall, On the cardinality of Lindelöf spaces with points \({ G}_{\delta }\), Topology and its Applications 63 (1995), 21–38.

    Google Scholar 

  180. F.D. Tall, Productively Lindelöf spaces may all be D, Canadian Mathematical Bulletin, tp appear.

    Google Scholar 

  181. F.D. Tall, Lindelöf spaces which are Menger, Hurewicz, Alster, productive or D, Topology and its Applications 158:18 (2011), 2556–2563.

    Google Scholar 

  182. F.D. Tall and B. Tsaban, On productively Lindelöf spaces, Topology and its Applications 158 (2011), 1239–1248.

    Google Scholar 

  183. S. Todorčević, Aronszajn orderings, Publications de l’Insitut Mathematique 57(71) (1995), 29–46.

    Google Scholar 

  184. B. Tsaban, The minimal cardinality where the Reznichenko property fails, Israel Journal of Mathematics 140 (2004), 367–374.

    Google Scholar 

  185. B. Tsaban, o-bounded groups and other topological groups with strong combinatorial properties, Proceedings of the American Mathematical Society 134 (2006), 881–891.

    Google Scholar 

  186. B. Tsaban, A new selection principle, Topology Proceedings 31 (2007), 319–329.

    Google Scholar 

  187. B. Tsaban and L. Zdomskyy Scales, fields, and a problem of Hurewicz, Journal of the European Mathematical Society 10 (2008), 837–866.

    Google Scholar 

  188. B. Tsaban and L. Zdomskyy, On the Pytkeev property in spaces of continuous functions (II), Houston Journal of Mathematics 35 (2009), 563–571.

    Google Scholar 

  189. B. Tsaban and L. Zdomskyy, Hereditarily Hurewicz spaces and Arhangel’skiĭ sheaf amalgamations, Journal of the European Mathematical Society 12 (2012), 353–372.

    Google Scholar 

  190. L.A. Tumarkin, Beitrag zur allgemeinen Dimensionstheorie, Mat. Sbornik 33 (1926), 57–86.

    Google Scholar 

  191. V.V. Uspenskij, On the spectrum of frequencies of function spaces, Moscow University Mathematics Bulletin 37 (1982), 40–45.

    Google Scholar 

  192. L. Zdomskyy, A semifilter approach to selection principles, Commentationes Mathematicae Universitatis Carolinae 46 (2005), 525–539.

    Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, Kanagawa University, Yokohama, 221-8686, Japan

    Masami Sakai

  2. Department of Mathematics, Boise State University, Boise, ID, 83725, USA

    Marion Scheepers

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  1. Masami Sakai
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  2. Marion Scheepers
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Correspondence to Masami Sakai .

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  1. Faculteit EWI, Technische Universiteit, Delft, The Netherlands

    K.P. Hart

  2. Faculty of Sciences, VU University, Amsterdam, The Netherlands

    J. van Mill

  3. Faculty of Mathematics and Physics, Charles University, Prague, Czech Republic

    P. Simon

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Sakai, M., Scheepers, M. (2014). The Combinatorics of Open Covers. In: Hart, K., van Mill, J., Simon, P. (eds) Recent Progress in General Topology III. Atlantis Press, Paris. https://doi.org/10.2991/978-94-6239-024-9_18

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