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Topology of Mrówka-Isbell Spaces

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Pseudocompact Topological Spaces

Part of the book series: Developments in Mathematics ((DEVM,volume 55))

Abstract

An infinite family \(\mathscr {A}\) of infinite subsets of the natural numbers, \(\omega \), is almost disjoint (AD) if the intersection of any two distinct elements of \(\mathscr {A}\) is finite. It is maximal almost disjoint (MAD) if given an infinite \(X\subset \omega \) there is an \(A\in \mathscr {A}\) such that \(|A\cap X|=\omega \), in other words, if the family \(\mathscr {A}\) is not included in any larger almost disjoint family.

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Notes

  1. 1.

    Recall that a separable metrizable space X is called a Q-set if every subset of X is \(G_\delta \) in X.

  2. 2.

    Remember that the bounding number, \(\mathfrak {b}\), is defined as the smallest cardinality of a family \(\mathscr {G}\subset \omega ^\omega \) which is \(\le ^*\)-unbounded

  3. 3.

    A partitioner P is trivial if \(P\in \mathscr {I_A}\) or \(\omega \setminus P\in \mathscr {I_A}\).

  4. 4.

    To see this let f be a function dominating all increasing enumerations \(e_\alpha \) of the sets \(A_\alpha \) and let \(g(0)=f(0)\) and \(g(n+1)=f(g(n)+1)\).

  5. 5.

    It is known that \(2^X\) is completely regular if and only if it is normal if and only if it is compact; hence we use the feeble compactness concept in the realm of the hyperspaces.

  6. 6.

    In fact, a more general result is valid, see Theorem 1.4.10.

  7. 7.

    A subset A of a topological space X is relatively countably compact in X if every \(E\in [A]^\omega \) has an accumulation point in X.

  8. 8.

    For the rest of this section we shall fix the following notation concerning weak selections. Given sets X and Y, and \(\psi : [X]^2\rightarrow X\) and \(\varphi : [Y]^2\rightarrow Y\) weak selections, we will say that \(\psi \) and \(\varphi \) are isomorphic, \(\psi \approx \varphi \), if there is a bijection \(\rho : X\rightarrow Y\) such that \(\psi (\{a,b\})=\varphi (\{\rho (a),\rho (b)\})\) for every \(a,b\in X\). We will also say that \(\psi \) is embedded in \(\varphi \) if \(\psi \approx \varphi \upharpoonright [A]^2\) for some \(A\subset X\). Let \(\varphi \) be a weak selection on a set X and let \(x,y\in X\). We will denote by \(x\rightarrow _\varphi y\) the condition \(\varphi (x,y)=y\). If \(A,B\subset X\), we will say that B dominates with A with respect to \(\varphi \), denoted by \(A\rightrightarrows _\varphi B\), if for every \(a\in A\) and \(b\in B\), \(a\rightarrow _\varphi b\). We will also say that A and B are aligned with respect to \(\varphi \) and denote by \(A||_{\varphi }B\), if \(A\rightrightarrows _\varphi B\) or \(B\rightrightarrows _\varphi A\). Given \(A,B\in [\omega ]^\omega \) and a weak selection \(\psi \) on \(\omega \), we will say that B almost dominates A with respect to \(\psi \) (or simply that B almost dominates A if \(\psi \) is clear from the context) and denote by \(A\rightrightarrows ^{*}_\psi B\), if there is a \(k\in \omega \) such that \(A\setminus k\rightrightarrows _\psi B\setminus k\). We will also say that A and B are almost aligned with respect to \(\psi \), denoted by \(A||^*_\psi B\), if \(A\rightrightarrows ^{*}_\psi B\) or \(B\rightrightarrows ^{*}_\psi A\). If \(n\in \omega \) then we will say that A is almost dominated by \(\{n\}\), which will be denoted by \(A\rightrightarrows ^*_\psi \{n\}\), whenever \(A\setminus k\rightrightarrows _\psi \{n\}\) for some \(k\in \omega \). In a similar way, we define \(\{n\}\rightrightarrows ^{*}_\psi A\) and \(\{n\}||^*_\psi A\). When the selection is clear from the context, we suppress the use of the subscript. Given a weak selection \(\varphi \), a triple \(\{a,b,c\}\) is called a 3-cycle if either \(a\rightarrow b\rightarrow c \rightarrow a\) or \(c\rightarrow b\rightarrow a\rightarrow c\).

  9. 9.

    Recall that a family \(\mathscr {I}\subset [\omega ]^\omega \) is independent if \(\bigcap \mathscr {F}\setminus \bigcup \mathscr {F'}\) is infinite for every \(\mathscr {F},\mathscr {F'}\) finite disjoint subsets of \(\mathscr {I}\).

  10. 10.

    To obtain such an independent family, start with an arbitrary independent family \(\mathscr {J}=\{J_n: n\in \omega \}\subset [\omega ]^\omega \) and recursively define a family \(\mathscr {I}=\{I_n: n\in \omega \}\) as follows:

    • \(I_0=J_0\);

    • \(I_{n+1}=(J_{n+1}\setminus \{k\le n:n+1\in I_k\})\cup \{k\le n:n+1\notin I_k\}\).

    For every \(n\in \omega \), the set \(I_n\in \mathscr {I}\) is obtained by finite changes of \(J_n\), guaranteeing that \(\mathscr {I}\) is also an independent family such that, \(n\in I_m\) if and only if \(m\not \in I_n\), for every \(n,m\in \omega \).

  11. 11.

    A space X is said to have property (a) provided for every open cover \(\mathcal U\) of X and every dense subset \(D\subset X\) there is a set F closed discrete in X contained in D such that \(st(F,\mathcal U)=X\), where \(st(F,\mathcal U)=\bigcup \{U\in \mathcal U: U\cap F\ne \emptyset \}\).

References

  1. U. Abraham, S. Shelah. Lusin sequences under CH and under Martin’s axiom. Fund. Math. 169 (2001), 97–103.

    Article  MathSciNet  Google Scholar 

  2. P. Alexandroff, P. Urysohn. Mémoire sur les espaces topologiques compacts. Verh. Akad. Wetensch. Amsterdam 14 (1926) 1–96.

    Google Scholar 

  3. M. Arciga-Alejandre. Extensiones continuas sobre espacios de Isbell-Mrówka, Master’s Thesis, Universidad Michoacana de San Nicolás de Hidalgo, Mexico (2007).

    Google Scholar 

  4. M. Arciga-Alejandre, M. Hrušák, C. Martínez-Ranero. Invariance properties of almost disjoint families. J. Symbolic Logic 78 (2013), 989–999.

    Article  MathSciNet  Google Scholar 

  5. A. V. Arhangel’skii, R. Z. Buzyakova. The rank of the diagonal and submetrizability. Comment. Math. Univ. Carolin. 47 (2006), 585–597.

    Google Scholar 

  6. A. V. Arhangel’skiĭ, S. P. Franklin. Ordinal invariants for topological spaces. Michigan Math. J. 15 (1968), 313–320.

    Google Scholar 

  7. G. Artico, U. Marconi, J. Pelant, L. Rotter, M. Tkachenko. Selections and suborderability. Fund. Math. 175 (2002), 1–33.

    Article  MathSciNet  Google Scholar 

  8. B. Balcar, J. Dočkálková, P. Simon. Almost disjoint families of countable sets. Finite and infinite sets, Vol. I, II (Eger, 1981), 59-88, Colloq. Math. Soc. János Bolyai, 37, North-Holland, Amsterdam, 1984.

    Chapter  Google Scholar 

  9. B. Balcar, J. Pelant, P. Simon. The space of ultrafilters on \({\bf N}\) covered by nowhere dense sets. Fund. Math. 110 (1) (1980) 11–24.

    Article  MathSciNet  Google Scholar 

  10. T. Bartoszyński, H. Judah. Set theory. On the structure of the real line. A K Peters, Ltd., 1985.

    Google Scholar 

  11. A. I. Bashkirov. The classification of quotient maps and sequential bicompacta. Soviet Math. Dokl. 15 (1974) 1104–1109.

    Google Scholar 

  12. A. I. Bashkirov. On continuous maps of Isbell spaces and strong \(0\)-dimensionality. Bull. Acad. Polon. Sci. Sr. Sci. Math. 27 (1979), 605–611 (1980).

    Google Scholar 

  13. A. I. Bashkirov. On Stone-Čech compactifications of Isbell spaces. Bull. Acad. Polon. Sci. Sr. Sci. Math. 27 (1979), 613–619 (1980).

    Google Scholar 

  14. J. E. Baumgartner, M. Weese. Partition algebras for almost-disjoint families. Trans. Amer. Math. Soc. 274 (1982), 619–630.

    Article  MathSciNet  Google Scholar 

  15. D. Bernal-Santos. The Rothberger property on \(C_p(\Psi ({\cal{A}}),2)\). Comment. Math. Univ. Carolin. 57 (2016), 83–88.

    Google Scholar 

  16. D. Bernal-Santos. The Rothberger property on \(C_p(X,2)\). Topology Appl. 196 (2015), part A, 106–119.

    Google Scholar 

  17. D. Bernal-Santos, Á. Tamariz-Mascarúa. The Menger property on \(C_p(X,2)\). Topology Appl. 183 (2015), 110–126.

    Google Scholar 

  18. A. Blass. Combinatorial cardinal characteristics of the continuum. In Handbook of set theory. Vols. 1, 2, 3, pages 395–489. Springer, Dordrecht, 2010.

    Google Scholar 

  19. J. Brendle, G. Piper. MAD families with strong combinatorial properties. Fund. Math. 193 (2007), 7–21.

    Article  MathSciNet  Google Scholar 

  20. R. Z. Buzyakova. In search for Lindelöf \(C_p\)’s. Comment. Math. Univ. Carolin. 45 (2004) 145–151.

    Google Scholar 

  21. J. Cao, T. Nogura, A. H. Tomita. Countable compactness of hyperspaces and Ginsburg’s questions. Topology Appl. 144 (2004) 133–145.

    Article  MathSciNet  Google Scholar 

  22. J. Chaber. Conditions which imply compactness in countably compact spaces. Bull. Acad. Pol. Sci. Ser. Math., 24 (1976) 993–998.

    Google Scholar 

  23. E. K. van Douwen. The integers and topology. Handbook of set-theoretic topology, 111–167, North-Holland, Amsterdam, 1984.

    Google Scholar 

  24. A. Dow. Dow’s questions. Open problems in topology, 5–11, North-Holland, Amsterdam, 1990.

    Google Scholar 

  25. A. Dow. Large compact separable spaces may all contain \(\beta {\mathbb{N}}\). Proc. Amer. Math. Soc. 109 (1990), 275–279.

    MathSciNet  MATH  Google Scholar 

  26. A. Dow. Sequential order under \({\sf {M}} A\). Topology Appl. 146-147 (2005) 501–510.

    Google Scholar 

  27. A. Dow. A non-partitionable mad family. Spring Topology and Dynamical Systems Conference. Topology Proc. 30 (2006), 181–186.

    Google Scholar 

  28. A. Dow. Sequential order under PFA. Canad. Math. Bull. 54 (2011) 270–276.

    Article  MathSciNet  Google Scholar 

  29. A. Dow. P. Simon. Spaces of continuous functions over a \(\Psi \)-space. Top. Appl. 153 (2006) 2260–2271.

    Google Scholar 

  30. A. Dow, J. E. Vaughan. Ordinal remainders of \(\psi \)-spaces. Topology Appl. 158 (2011), 1852–1857.

    Google Scholar 

  31. A. Dow, J. Vaughan. Ordinal remainders of classical \(\psi \)-spaces. Fund. Math. 217 (2012) 83–93.

    Article  MathSciNet  Google Scholar 

  32. A. Dow, J. E. Vaughan. Mrówka maximal almost disjoint families for uncountable cardinals. Topology Appl. 157 (2010), 1379–1394.

    Article  MathSciNet  Google Scholar 

  33. R. Engelking. General topology, second ed. Sigma Series in Pure Mathematics, vol. 6, Heldermann Verlag, Berlin, 1989. Translated from the Polish by the author.

    Google Scholar 

  34. I. Farah. Luzin gaps. Trans. Amer. Math. Soc. 356 (2004) 2197–2239.

    Article  MathSciNet  Google Scholar 

  35. J. Ferrer. On the Controlled Separable Projection Property for some C(K) spaces. Acta Math. Hung. 124 (2009), 145–154.

    Article  MathSciNet  Google Scholar 

  36. J. Ferrer, P. Koszmider and W. Kubiś. Almost disjoint families of countable sets and separable complementation properties. J. Math. Anal. Appl. 401 (2013), 939–949.

    Article  MathSciNet  Google Scholar 

  37. S. García-Ferreira. Continuous functions between Isbell-Mrówka spaces. Comment. Math. Univ. Carolin. 39 (1998), 185–195.

    Google Scholar 

  38. S. García-Ferreira, M. Sanchis. Weak selections and pseudocompactness. Proc. Amer. Math. Soc. 132 (2004) 1823–1825.

    Article  MathSciNet  Google Scholar 

  39. M. A. Gaspar, F. Hernández-Hernández, M. Hrušák. Scattered spaces from weak diamonds. To appear in Israel J. Math. (2018).

    Google Scholar 

  40. S. Ghasemi, P. Koszmider. An extension of compact operators by compact operators with no nontrivial multipliers. Preprint (2017).

    Google Scholar 

  41. L. Gillman, M. Jerison. Rings of Continuous Functions. Van Nostrand, Princeton, NJ. 1960.

    Chapter  Google Scholar 

  42. J. Ginsburg. Some results on the countable compactness and pseudocompactness of hyperspaces. Canad. J. Math. 27 (1975) 1392–1399.

    Article  MathSciNet  Google Scholar 

  43. I. Glicksberg. Stone-Čech compactifications of products. Trans. Amer. Math. Soc. 90 (1959) 369–382.

    MathSciNet  MATH  Google Scholar 

  44. V. Gutev, T. Nogura. Selection problems for hyperspaces. In Open Problems in Topology II, E. Pearl, editor. Elsevier B. V. 2007.

    Chapter  Google Scholar 

  45. O. Guzmán, M. Hrušák. \(n\)-Luzin gaps. Top. appl. 160 (2013) 1364–1374.

    Google Scholar 

  46. M. Hrušák. Almost disjoint families and topology. Recent progress in general topology. III, 601-638, Atlantis Press, Paris, 2014.

    Google Scholar 

  47. M. Hrušák, S. García-Ferreira. Ordering MAD families a la Katětov. J. Symbolic Logic 68 (2003), 1337–1353.

    Article  MathSciNet  Google Scholar 

  48. M. Hrušák, F. Hernández-Hernández, I. Martínez-Ruiz. Pseudocompactness of hyperspaces. Topol. Appl. 154 (2007) 3048–3055.

    Google Scholar 

  49. M. Hrušák, I. Martínez-Ruiz. Selections and weak orderability. Fund. Math. 203 (2009) 1–20.

    Article  MathSciNet  Google Scholar 

  50. M. Hrušák, C. J. D. Morgan, S. G. da Silva. Luzin gaps are not countably paracompact. Questions Answers Gen. Topology 30 (2012), 59–66.

    Google Scholar 

  51. M. Hrušák, R. Raphael, R. G. Woods. On a class of pseudocompact spaces derived from ring epimorphisms. Topology Appl. 153 (2005), 541–556.

    Article  MathSciNet  Google Scholar 

  52. M. Hrušák, P. J. Szeptycki, Á. Tamariz-Mascarúa. Spaces of continuous functions defined on Mrówka spaces. Top. Appl. 148 (2005) 239–252.

    Article  MathSciNet  Google Scholar 

  53. M. Hrušák, P. J. Szeptycki, A. H. Tomita. Selections on \(\Psi \)-spaces. Comment Math. Univ. Carolin. 42 (2001) 763–769.

    Google Scholar 

  54. W. Just, O. V. Sipacheva, P. J. Szeptycki. Nonnormal spaces \(C_p(X)\) with countable extent. Proc. Amer. Math. Soc. 124 (1996), 1227–1235.

    Article  MathSciNet  Google Scholar 

  55. V. Kannan, M. Rajagopalan. Hereditarily locally compact separable spaces. In Categorical topology (Proc. Internat. Conf. Free Univ. Berlin). Lecture Notes in Math. Vol. 71 pages 185-195. Springer, Berlin 1979.

    Google Scholar 

  56. P. Koszmider. On decompositions of Banach spaces of continuous functions on Mrówka’s spaces. Proc. Amer. Math. Soc. 133 (2005), 2137–2146.

    Article  MathSciNet  Google Scholar 

  57. J. Kulesza, R. Levy. Separation in \(\Psi \)-spaces. Topology Appl. 42 (1991), 101–107.

    Google Scholar 

  58. K. Kunen. Set theory: an introduction to independence proofs, vol. 102, North-Holland Publishing Co., Amsterdam, 1980.

    Google Scholar 

  59. N. N. Luzin. On subsets of the series of natural numbers. Izvestiya Akad. Nauk SSSR. Ser. Mat. 11 (1947) 403–410.

    Google Scholar 

  60. V. I. Malykhin, Á. Tamariz-Mascarúa. Extensions of functions in Mrówka-Isbell spaces. Topology Appl. 81 (1997), 85–102.

    Article  MathSciNet  Google Scholar 

  61. A. R. D. Mathias. Happy families. Annals of Mathematical Logic 12 (1977) 59–111.

    Article  MathSciNet  Google Scholar 

  62. W. Marciszewski. A function space \(C(K)\) not weakly homeomorphic to \(C(K)\times C(K)\). Studia Math. 88 (1988), 129–137.

    Article  MathSciNet  Google Scholar 

  63. M. V. Matveev. Some questions on property (a). Questions Answers Gen. Topology 15 (1997), 103–111.

    Google Scholar 

  64. E. Michael. Topologies on spaces of subsets. Trans. Amer. Math. Soc. 71 (1951) 152–182.

    Article  MathSciNet  Google Scholar 

  65. J. van Mill, E. Wattel. Selections and orderability. Proc. Amer. Math. Soc. 83 (1981) 601–605.

    Article  MathSciNet  Google Scholar 

  66. A. W. Miller. A MAD Q-set. Fund. Math. 178 (2003), 271–281.

    Article  MathSciNet  Google Scholar 

  67. J. T. Moore, M. Hrušák, M. Džamonja. Parametrized \(\diamondsuit \) principles. Trans. Amer. Math. Soc. 356 (2004), 2281–2306.

    Article  MathSciNet  Google Scholar 

  68. C. J. G. Morgan, S. G. da Silva. Almost disjoint families and “never” cardinal invariants. Comment. Math. Univ. Carolin. 50 (2009), 433–444.

    Google Scholar 

  69. C. J. G. Morgan, S. G. da Silva. Selectively (a)-spaces from almost disjoint families are necessarily countable under a certain parametrized weak diamond principle. Houston J. Math. 42 (2016), 1031–1046.

    Google Scholar 

  70. S. Mrówka. On completely regular spaces. Fund. Math. 41 (1954) 105–106.

    Article  MathSciNet  Google Scholar 

  71. S. Mrówka. Some set-theoretic constructions in topology. Fund. Math., 94, (1977) 83–92.

    Article  MathSciNet  Google Scholar 

  72. J. Novák. On the Cartesian product of two compact spaces. Fund. Math. 40 (1953) 106–112.

    Article  MathSciNet  Google Scholar 

  73. J. Roitman, L. Soukup. Luzin and anti-Luzin almost disjoint families. Fund. Math. 158 (1998), 51–67.

    Google Scholar 

  74. S. G. da Silva. On the extent of separable, locally compact, selectively (a)-spaces. Colloq. Math. 141 (2015), 199–208.

    Article  MathSciNet  Google Scholar 

  75. S. G. da Silva. (a)-spaces and selectively (a)-spaces from almost disjoint families. Acta Math. Hungar. 142 (2014), 420–432.

    Google Scholar 

  76. S. G. da Silva. On the presence of countable paracompactness, normality and property (a) in spaces from almost disjoint families. Questions Answers Gen. Topology 25 (2007), 1–18.

    Google Scholar 

  77. P. Simon. A compact Fréchet space whose square is not Fréchet. Comment. Math. Univ. Carolin. 21 (1980) 749–753.

    Google Scholar 

  78. J. Steprāns. Combinatorial consequences of adding Cohen reals. Set theory of the reals (Ramat Gan, 1991), 583–617, Israel Math. Conf. Proc., 6, Bar-Ilan Univ., Ramat Gan, 1993.

    Google Scholar 

  79. P. J. Szeptycki. Soft almost disjoint families. Proc. Amer. Math. Soc. 130 (2002), 3713–3717.

    Google Scholar 

  80. P. J. Szeptycki. J. E. Vaughan. Almost disjoint families and property (a). Fund. Math. 158 (1998), 229–240.

    Google Scholar 

  81. P. Szeptycki. Countable metacompactness in \(\Psi \)-spaces. Proc. Amer. Math. Soc. 120 (1994), 1241–1246.

    MathSciNet  MATH  Google Scholar 

  82. F. D. Tall. Set-Theoretic Consistency Results and Topological Theorems Concerning The Normal Moore Space Conjecture and Related Problems. Thesis (Ph.D.) The University of Wisconsin - Madison. 1969. 105 pp.

    Google Scholar 

  83. H. Terasaka. On Cartesian products of compact spaces. Osaka Math. J. 4 (1952) 11–15.

    Google Scholar 

  84. J. Terasawa. Spaces \(N\cup \mathscr {R}\) and their dimensions. Topology Appl. 11 (1980) 93–102.

    Google Scholar 

  85. S. Todorčević. Analytic gaps. Fund. Math. 150 (1996) 55–66.

    Google Scholar 

  86. J. E. Vaughan, C. Payne. Fibers of continuous real-valued functions on \(\psi \)-spaces. Topology Appl. 195 (2015), 256–264.

    Google Scholar 

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Authors and Affiliations

  1. Facultad de Ciencias Físico Matemáticas, Universidad Michoacana de San Nicolás de Hidalgo, 58060, Morelia, Michoacán, México

    F. Hernández-Hernández

  2. Centro de Ciencias Matemáticas, Universidad Nacional Autónoma de México, 58089, Morelia, Michoacán, Mexico

    M. Hrušák

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  1. Centro de Ciencias Matemáticas, Universidad Nacional Autónoma de México-Campus Morelia, Morelia, Michoacán, Mexico

    Michael Hrušák

  2. Departamento de Matemáticas, Facultad de Ciencias, Universidad Nacional Autónoma de México, Ciudad de México, Mexico

    Ángel Tamariz-Mascarúa

  3. Departamento de Matemáticas, Universidad Autónoma Metropolitana, Ciudad de México, Mexico

    Mikhail Tkachenko

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Hernández-Hernández, F., Hrušák, M. (2018). Topology of Mrówka-Isbell Spaces. In: Hrušák, M., Tamariz-Mascarúa, Á., Tkachenko, M. (eds) Pseudocompact Topological Spaces. Developments in Mathematics, vol 55. Springer, Cham. https://doi.org/10.1007/978-3-319-91680-4_8

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