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Combinatorial set theory and cardinal function inequalities


Author: R. E. Hodel
Journal: Proc. Amer. Math. Soc. 111 (1991), 567-575
MSC: Primary 54A25; Secondary 04A20
DOI: https://doi.org/10.1090/S0002-9939-1991-1039531-7
MathSciNet review: 1039531
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Abstract: Three theorems of combinatorial set theory are proven. From the first we obtain the de Groot inequality $ \left\vert X \right\vert \leq {2^{hL(X)}}$, the Ginsburg-Woods inequality $ \left\vert X \right\vert \leq {2^{e(X)\Delta (X)}}$, the Erdös-Rado Partition Theorem for $ n = 2$, and set-theoretic versions of the Hajnal-Juhász inequalities $ \left\vert X \right\vert \leq {2^{c(X)\chi (X)}}$ and $ \left\vert X \right\vert \leq {2^{s(X)\psi (X)}}$. From the second we obtain a generalization of the Arhangel'skiĭ inequality $ \left\vert X \right\vert \leq {2^{L(X)\chi (X)}}$. From the third we obtain the Charlesworth inequality $ n(X) \leq psw{(X)^{L(X)}}$ and a generalization of the Burke-Hodel inequality $ \left\vert {K(X)} \right\vert \leq {2^{e(X)psw(X)}}$.


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  • [A] A. V. Arhangel'skiĭ, On the cardinality of bicompacta satisfying the axiom of first countability, Soviet Math. Dokl. 10 (1969), 951-955.
  • [BH] D. K. Burke and R. E. Hodel, The number of compact subsets of a topological space, Proc. Amer. Math. Soc. 58 (1976), 363-368. MR 0418014 (54:6058)
  • [C] A. Charlesworth, On the cardinality of a topological space, Proc. Amer. Math. Soc. 66 (1977), 138-142. MR 0451184 (56:9471)
  • [E] R. Engelking, General topology, Heldermann, Berlin, 1989. MR 1039321 (91c:54001)
  • [EK] R. Engelking and M. Karlowicz, Some theorems of set theory and their topological consequences, Fund. Math. 57 (1965), 275-285. MR 0196693 (33:4880)
  • [ER1] P. Erdös and R. Rado, A partition calculus in set theory, Bull. Amer. Math. Soc. 62 (1956), 427-489. MR 0081864 (18:458a)
  • [ER2] -, Intersection theorems for systems of sets, J. London Math. Soc. 35 (1960), 85-90. MR 0111692 (22:2554)
  • [GW] J. Ginsburg and G. Woods, A cardinal inequality for topological spaces involving closed discrete sets, Proc. Amer. Math. Soc. 64 (1977), 357-360. MR 0461407 (57:1392)
  • [G] J. de Groot, Discrete subspaces of Hausdorff spaces, Bull. Acad. Polon. Sci. 13 (1965), 537-544. MR 0210061 (35:956)
  • [HJ] A. Hajnal and I. Juhász, Discrete subsets of topological spaces, Indag. Math. 29 (1967), 343-356. MR 0229195 (37:4769)
  • [H1] R. Hodel, On a theorem of Arhangel'skiĭ concerning Lindelöf $ p$-spaces, Canad. J. Math. 27 (1975), 459-468. MR 0375205 (51:11401)
  • [H2] -, A technique for proving inequalities in cardinal functions, Topology Proc. 4 (1979), 115-120. MR 583694 (81m:54012)
  • [H3] -, Cardinal functions. I, in Handbook of Set-Theoretic Topology (K. Kunen and J. E. Vaughan, eds.), North-Holland, Amsterdam, 1984, pp. 1-61. MR 776620 (86j:54007)
  • [J] I. Juhász, Cardinal functions in topology--ten years later, Mathematisch Centrum, Amsterdam, 1980.
  • [M] E. Michael, A note on intersections, Proc. Amer. Math. Soc. 13 (1962), 281-283. MR 0133236 (24:A3070)
  • [Mi] A. Miščenko, Spaces with point-countable bases, Soviet Math. Dokl. 3 (1962), 855-858.
  • [MM] Dai MuMing, A topological space cardinality inequality involving the *Lindelöf number, Acta Math. Sinica 26 (1983), 731-735.
  • [P] R. Pol, Short proofs of two theorems on cardinality of topological spaces, Bull. Acad. Polon. Sci. 22 (1974), 1245-1249. MR 0383333 (52:4214)
  • [S] B. Šapirovskiĭ, On discrete subspaces of topological spaces. Weight, tightness and Souslin number, Soviet Math. Dokl. 13 (1972), 215-219.
  • [SW] S. H. Sun and Y. M. Wang, A strengthened topological cardinal inequality, Bull. Austral. Math. Soc. 32 (1985), 375-378. MR 819569 (87d:54009)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1991-1039531-7
Keywords: Lindelöf degree, extent, separating cover, Erdös-Rado Partition Theorem, cardinal functions
Article copyright: © Copyright 1991 American Mathematical Society

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