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Antidiamond principles and topological applications


Authors: Todd Eisworth and Peter Nyikos
Journal: Trans. Amer. Math. Soc. 361 (2009), 5695-5719
MSC (2000): Primary 03E75
DOI: https://doi.org/10.1090/S0002-9947-09-04705-9
Published electronically: June 24, 2009
MathSciNet review: 2529910
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Abstract: We investigate some combinatorial statements that are strong enough to imply that $ \diamondsuit$ fails (hence the name antidiamonds); yet most of them are also compatible with CH. We prove that these axioms have many consequences in set-theoretic topology, including the consistency, modulo large cardinals, of a Yes answer to a problem on linearly Lindelöf spaces posed by Arhangel'skiı and Buzyakova (1998).


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  • 1. A. V. Arhangel'skiı and R. Z. Buzyakova, Convergence in compacta and linear Lindelöfness, Comment. Math. Univ. Carolinae 39, 1 (1998) 159-166. MR 1623006 (99c:54034)
  • 2. U. Abraham and S. Todorčević, Consistency with CH of a combinatorial principle that implies there are no Suslin trees and all $ (\omega_1, \omega_1)$-gaps are Hausdorff, preprint.
  • 3. -, Partition properties of $ \omega_1$ compatible with CH, Fund. Math. 152 (1997) 165-181. MR 1441232 (98b:03064)
  • 4. K. Devlin and S. Shelah, Suslin properties and tree topologies, Proc. London Math. Soc. 39 (1979) 237-252. MR 548979 (80m:54031)
  • 5. E. K. van Douwen, The integers and topology, in: Handbook of Set-Theoretic Topology, K. Kunen and J. Vaughan ed., North-Holland (1984) 111-167. MR 776619 (85k:54001)
  • 6. T. Eisworth, CH and first countable, countably compact spaces, Topology Appl. 109 (2001), no. 1, 55-73. MR 1804563 (2001k:54010)
  • 7. -, Totally proper forcing and the Moore-Mrówka problem, Fund. Math. 177 (2003), no. 2, 121-136. MR 1992528 (2004g:03093)
  • 8. T. Eisworth and P. Nyikos, First countable, countably compact spaces and the continuum hypothesis, Trans. Amer. Math. Soc. 357 (2005), 4269-4299. MR 2156711 (2006g:03091)
  • 9. T. Eisworth and J. Roitman, CH with no Ostaszewski spaces, Trans. Amer. Math. Soc. 351 (1999), no. 7, 2675-2693. MR 1638230 (2000b:03182)
  • 10. R. Engelking, General Topology, Heldermann-Verlag, Berlin, 1989. MR 1039321 (91c:54001)
  • 11. G. Gruenhage, Some results on spaces having an orthobase or a base of subinfinite rank, Proceedings of the 1977 Topology Conference (Louisiana State Univ., Baton Rouge, La., 1977) I. Topology Proc. 2 (1977), no. 1, 151-159. MR 540602 (80k:54056)
  • 12. G. Gruenhage, Paracompactness and subparacompactness in perfectly normal locally compact spaces, Usp. Mat. Nauk 35, no. 3 (1980) 44-49 (Russian) International Topology Conference (Moscow State Univ., Moscow, 1979). Translated from the English by A. V. Arhangel'skiı. MR 580619 (81k:54021)
  • 13. N. Howes, A note on transfinite sequences, Fund. Math. 106 (1980) 213-216. MR 584493 (82c:54002)
  • 14. J. Hirschorn, Random trees under CH, Israel J. Math. 157 (2007) 123-154. MR 2342443 (2008g:03084)
  • 15. Tetsuya Ishiu, $ \alpha$-properness and Axiom A, Fund. Math. 186 (2005), no. 1, 25-37. MR 2163100 (2006g:03081)
  • 16. N. N. Jakovlev, On the theory of o-metrizable spaces, Dokl. Akad. Nauk. SSSR 229 (1976) 1330-1331. MR 0415578 (54:3663)
  • 17. I. Juhász, Consistency results in topology, in: Handbook of Mathematical Logic, J. Barwise ed., North-Holland, 1977, 503-522. MR 0457132 (56:15351)
  • 18. I. Juhász, L. Soukup, and S. Szentmiklóssy, What is left of CH after you add Cohen reals? Top. Appl. 85 (1998) 165-174. MR 1617461 (99d:54001)
  • 19. K. Kunen, Locally compact linearly Lindelöf spaces, Comment. Math. Univ. Carolinae 43 (2002) 155-158. MR 1903314 (2003d:54040)
  • 20. P. J. Nyikos, Subsets of $ {}^\omega\omega$ and the Fréchet-Urysohn property, Top. Appl. 48 (1992) 91-116. MR 1195504 (93k:54011)
  • 21. -, Various topologies on trees, in: Proceedings of the Tennessee Topology Conference, P. R. Misra and M. Rajagopalan, eds., World Scientific Publishing Co., 1997, 167-198. MR 1607401 (98m:54037)
  • 22. -, The theory of nonmetrizable manifolds, in: Handbook of Set-Theoretic Topology, K. Kunen and J. Vaughan ed., North-Holland, 1984, 633-684. MR 776619 (85k:54001)
  • 23. -, Applications of some strong set-theoretic axioms to locally compact $ T_5$ and hereditarily scwH spaces, Fund. Math. 176 (1) (2003) 25-45. MR 1971471 (2004k:54008)
  • 24. -, Hereditarily normal, locally compact Dowker spaces, Topology Proceedings 24 (1999) 261-276. MR 1802691 (2001i:54017)
  • 25. -, Correction to ``Complete normality and metrization theory of manifolds,'' Top. Appl. 138 (2004) 325-327. MR 2035491 (2004k:54030)
  • 26. -, Applications of antidiamond and anti-PFA axioms to metrization of manifolds, preprint.
  • 27. T. Przymusinski, Products of normal spaces, in: Handbook of Set-Theoretic Topology, K. Kunen and J. Vaughan ed., North-Holland, 1984, 781-826. MR 776637 (86c:54007)
  • 28. E. Pearl, Linearly Lindelöf problems, in: Open Problems in Topology II, E. Pearl, ed., Elsevier B.V., 2007, 225-231. MR 2367385
  • 29. J. Roitman, Basic S and L, in: Handbook of Set-Theoretic Topology, K. Kunen and J. Vaughan, ed., North-Holland, 1984, pp. 295-326. MR 776626 (87a:54043)
  • 30. M. E. Rudin, Lectures on Set Theoretic Topology, CBMS Regional Conference Series No. 23, American Mathematical Society, Providence, RI, 1977. MR 0367886 (51:4128)
  • 31. -, The undecidability of the existence of a perfectly normal nonmetrizable manifold, Houston J. Math. 5 (1979) 105-112. MR 546759 (80j:54014)
  • 32. -, A nonmetrizable manifold from $ \diamondsuit^+$, Top. Appl. 28 (1988) 105-112. MR 932975 (89c:54049)
  • 33. M. E. Rudin and P. L. Zenor, A perfectly normal nonmetrizable manifold, Houston Math. J. 2 (1976) 129-134. MR 0394560 (52:15361)
  • 34. S. Shelah, Proper and improper forcing, Perspectives in Mathematical Logic, Springer, 1998. MR 1623206 (98m:03002)
  • 35. C. Schlindwein, Suslin's hypothesis does not imply stationary antichains, Ann. Pure and Appl. Logic 32 (1993) 153-167. MR 1241252 (94i:03101)
  • 36. -, SH plus CH does not imply stationary antichains, Ann. Pure and Appl. Logic 124 (2003) 233-265. MR 2013399 (2004j:03056)
  • 37. S. Todorčević, A dichotomy for P-ideals of countable sets, Fund. Math. 166 (2000) 251-267. MR 1809418 (2001k:03111)

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Additional Information

Todd Eisworth
Affiliation: Department of Mathematics, University of Northern Iowa. Cedar Falls, Iowa 50614
Email: eisworth@math.uni.edu

Peter Nyikos
Affiliation: Department of Mathematics, University of South Carolina, Columbia, South Carolina
Email: nyikos@math.sc.edu

DOI: https://doi.org/10.1090/S0002-9947-09-04705-9
Keywords: Diamond, continuum hypothesis, forcing, S-space, P-ideal, antidiamond
Received by editor(s): July 14, 2005
Received by editor(s) in revised form: May 4, 2007
Published electronically: June 24, 2009
Additional Notes: The first author was partially supported by a University of Northern Iowa Summer Fellowship and NSF Grant DMS-0506063
The research of the second author was partially supported by NSF Grant DMS-9322613.
Article copyright: © Copyright 2009 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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