Antidiamond principles and topological applications

Eisworth, Todd; Nyikos, Peter

doi:10.1090/S0002-9947-09-04705-9

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
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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).

1. A. V. Arhangel′skii and R. Z. Buzyakova, Convergence in compacta and linear Lindelöfness, Comment. Math. Univ. Carolin. 39 (1998), no. 1, 159–166. MR 1623006
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. Uri Abraham and Stevo Todorčević, Partition properties of 𝜔₁ compatible with CH, Fund. Math. 152 (1997), no. 2, 165–181. MR 1441232
4. Keith J. Devlin and Saharon Shelah, Souslin properties and tree topologies, Proc. London Math. Soc. (3) 39 (1979), no. 2, 237–252. MR 548979, https://doi.org/10.1112/plms/s3-39.2.237
5. Kenneth Kunen and Jerry E. Vaughan (eds.), Handbook of set-theoretic topology, North-Holland Publishing Co., Amsterdam, 1984. MR 776619
6. Todd Eisworth, CH and first countable, countably compact spaces, Topology Appl. 109 (2001), no. 1, 55–73. MR 1804563, https://doi.org/10.1016/S0166-8641(99)00126-1
7. Todd Eisworth, Totally proper forcing and the Moore-Mrówka problem, Fund. Math. 177 (2003), no. 2, 121–137. MR 1992528, https://doi.org/10.4064/fm177-2-2
8. Todd Eisworth and Peter Nyikos, First countable, countably compact spaces and the continuum hypothesis, Trans. Amer. Math. Soc. 357 (2005), no. 11, 4269–4299. MR 2156711, https://doi.org/10.1090/S0002-9947-05-04034-1
9. Todd Eisworth and Judith Roitman, 𝐶𝐻 with no Ostaszewski spaces, Trans. Amer. Math. Soc. 351 (1999), no. 7, 2675–2693. MR 1638230, https://doi.org/10.1090/S0002-9947-99-02407-1
10. Ryszard Engelking, General topology, 2nd ed., Sigma Series in Pure Mathematics, vol. 6, Heldermann Verlag, Berlin, 1989. Translated from the Polish by the author. MR 1039321
11. Gary 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, 1977, pp. 151–159 (1978). MR 540602
12. G. Grjunhage, Paracompactness and subparacompactness in perfectly normal locally bicompact spaces, Uspekhi Mat. Nauk 35 (1980), no. 3(213), 44–49 (Russian). International Topology Conference (Moscow State Univ., Moscow, 1979); Translated from the English by A. V. Arhangel′skiĭ. MR 580619
13. Norman R. Howes, A note on transfinite sequences, Fund. Math. 106 (1980), no. 3, 213–226. MR 584493, https://doi.org/10.4064/fm-106-3-213-226
14. James Hirschorn, Random trees under CH, Israel J. Math. 157 (2007), 123–153. MR 2342443, https://doi.org/10.1007/s11856-006-0005-3
15. Tetsuya Ishiu, 𝛼-properness and Axiom A, Fund. Math. 186 (2005), no. 1, 25–37. MR 2163100, https://doi.org/10.4064/fm186-1-2
16. N. N. Jakovlev, On the theory of 𝑜-metrizable spaces, Dokl. Akad. Nauk SSSR 229 (1976), no. 6, 1330–1331 (Russian). MR 0415578
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 Z. Szentmiklóssy, What is left of CH after you add Cohen reals?, Topology Appl. 85 (1998), no. 1-3, 165–174. 8th Prague Topological Symposium on General Topology and Its Relations to Modern Analysis and Algebra (1996). MR 1617461, https://doi.org/10.1016/S0166-8641(97)00148-X
19. Kenneth Kunen, Locally compact linearly Lindelöf spaces, Comment. Math. Univ. Carolin. 43 (2002), no. 1, 155–158. MR 1903314
20. Peter J. Nyikos, Subsets of ^{𝜔}𝜔 and the Fréchet-Urysohn and 𝛼ᵢ-properties, Topology Appl. 48 (1992), no. 2, 91–116. MR 1195504, https://doi.org/10.1016/0166-8641(92)90021-Q
21. Peter J. Nyikos, Various topologies on trees, Proceedings of the Tennessee Topology Conference (Nashville, TN, 1996) World Sci. Publ., River Edge, NJ, 1997, pp. 167–198. MR 1607401
22. Kenneth Kunen and Jerry E. Vaughan (eds.), Handbook of set-theoretic topology, North-Holland Publishing Co., Amsterdam, 1984. MR 776619
23. Peter J. Nyikos, Applications of some strong set-theoretic axioms to locally compact 𝑇₅ and hereditarily scwH spaces, Fund. Math. 176 (2003), no. 1, 25–45. MR 1971471, https://doi.org/10.4064/fm176-1-3
24. Peter J. Nyikos, Hereditarily normal, locally compact Dowker spaces, Proceedings of the 1999 Topology and Dynamics Conference (Salt Lake City, UT), 1999, pp. 261–276. MR 1802691
25. Peter Nyikos, Correction to: “Complete normality and metrization theory of manifolds” [Topology Appl. 123 (2002), no. 1 181–192; MR1921659], Topology Appl. 138 (2004), no. 1-3, 325–327. MR 2035491, https://doi.org/10.1016/j.topol.2003.11.004
26. -, Applications of antidiamond and anti-PFA axioms to metrization of manifolds, preprint.
27. Teodor C. Przymusiński, Products of normal spaces, Handbook of set-theoretic topology, North-Holland, Amsterdam, 1984, pp. 781–826. MR 776637
28. Elliott Pearl (ed.), Open problems in topology. II, Elsevier B. V., Amsterdam, 2007. MR 2367385
29. Judy Roitman, Basic 𝑆 and 𝐿, Handbook of set-theoretic topology, North-Holland, Amsterdam, 1984, pp. 295–326. MR 776626
30. Mary Ellen Rudin, Lectures on set theoretic topology, Published for the Conference Board of the Mathematical Sciences by the American Mathematical Society, Providence, R.I., 1975. Expository lectures from the CBMS Regional Conference held at the University of Wyoming, Laramie, Wyo., August 12–16, 1974; Conference Board of the Mathematical Sciences Regional Conference Series in Mathematics, No. 23. MR 0367886
31. Mary Ellen Rudin, The undecidability of the existence of a perfectly normal nonmetrizable manifold, Houston J. Math. 5 (1979), no. 2, 249–252. MR 546759
32. Mary Ellen Rudin, A nonmetrizable manifold from \lozenge⁺, Topology Appl. 28 (1988), no. 2, 105–112. Special issue on set-theoretic topology. MR 932975, https://doi.org/10.1016/0166-8641(88)90002-8
33. Mary Ellen Rudin and Phillip Zenor, A perfectly normal nonmetrizable manifold, Houston J. Math. 2 (1976), no. 1, 129–134. MR 0394560
34. Saharon Shelah, Proper and improper forcing, 2nd ed., Perspectives in Mathematical Logic, Springer-Verlag, Berlin, 1998. MR 1623206
35. Chaz Schlindwein, Suslin’s hypothesis does not imply stationary antichains, Ann. Pure Appl. Logic 64 (1993), no. 2, 153–167. MR 1241252, https://doi.org/10.1016/0168-0072(93)90032-9
36. Chaz Schlindwein, SH plus CH does not imply stationary antichains, Ann. Pure Appl. Logic 124 (2003), no. 1-3, 233–265. MR 2013399, https://doi.org/10.1016/S0168-0072(03)00057-5
37. Stevo Todorčević, A dichotomy for P-ideals of countable sets, Fund. Math. 166 (2000), no. 3, 251–267. MR 1809418