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 Free Access

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’skiĭ and Buzyakova (1998).

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

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.

Uri Abraham and Stevo Todorčević, Partition properties of $\omega _1$ compatible with CH, Fund. Math. 152 (1997), no. 2, 165–181. MR 1441232, DOI https://doi.org/10.4064/fm-152-2-165-181

Keith J. Devlin and Saharon Shelah, Souslin properties and tree topologies, Proc. London Math. Soc. (3) 39 (1979), no. 2, 237–252. MR 548979, DOI https://doi.org/10.1112/plms/s3-39.2.237

Kenneth Kunen and Jerry E. Vaughan (eds.), Handbook of set-theoretic topology, North-Holland Publishing Co., Amsterdam, 1984. MR 776619

Todd Eisworth, CH and first countable, countably compact spaces, Topology Appl. 109 (2001), no. 1, 55–73. MR 1804563, DOI https://doi.org/10.1016/S0166-8641%2899%2900126-1

Todd Eisworth, Totally proper forcing and the Moore-Mrówka problem, Fund. Math. 177 (2003), no. 2, 121–137. MR 1992528, DOI https://doi.org/10.4064/fm177-2-2

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, DOI https://doi.org/10.1090/S0002-9947-05-04034-1

Todd Eisworth and Judith Roitman, ${\rm CH}$ with no Ostaszewski spaces, Trans. Amer. Math. Soc. 351 (1999), no. 7, 2675–2693. MR 1638230, DOI https://doi.org/10.1090/S0002-9947-99-02407-1

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

Gary Gruenhage, Some results on spaces having an orthobase or a base of subinfinite rank, Topology Proc. 2 (1977), no. 1, 151–159 (1978). MR 540602

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

Norman R. Howes, A note on transfinite sequences, Fund. Math. 106 (1980), no. 3, 213–226. MR 584493, DOI https://doi.org/10.4064/fm-106-3-213-226

James Hirschorn, Random trees under CH, Israel J. Math. 157 (2007), 123–153. MR 2342443, DOI https://doi.org/10.1007/s11856-006-0005-3

Tetsuya Ishiu, $\alpha $-properness and Axiom A, Fund. Math. 186 (2005), no. 1, 25–37. MR 2163100, DOI https://doi.org/10.4064/fm186-1-2

N. N. Jakovlev, On the theory of $o$-metrizable spaces, Dokl. Akad. Nauk SSSR 229 (1976), no. 6, 1330–1331 (Russian). MR 0415578

Handbook of mathematical logic, Studies in Logic and the Foundations of Mathematics, vol. 90, North-Holland Publishing Co., Amsterdam, 1977. With the cooperation of H. J. Keisler, K. Kunen, Y. N. Moschovakis and A. S. Troelstra. MR 457132

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, DOI https://doi.org/10.1016/S0166-8641%2897%2900148-X

Kenneth Kunen, Locally compact linearly Lindelöf spaces, Comment. Math. Univ. Carolin. 43 (2002), no. 1, 155–158. MR 1903314

Peter J. Nyikos, Subsets of ${}^\omega \omega $ and the Fréchet-Urysohn and $\alpha _i$-properties, Topology Appl. 48 (1992), no. 2, 91–116. MR 1195504, DOI https://doi.org/10.1016/0166-8641%2892%2990021-Q

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

Kenneth Kunen and Jerry E. Vaughan (eds.), Handbook of set-theoretic topology, North-Holland Publishing Co., Amsterdam, 1984. MR 776619

Peter J. Nyikos, Applications of some strong set-theoretic axioms to locally compact $T_5$ and hereditarily scwH spaces, Fund. Math. 176 (2003), no. 1, 25–45. MR 1971471, DOI https://doi.org/10.4064/fm176-1-3

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

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, DOI https://doi.org/10.1016/j.topol.2003.11.004

---, Applications of antidiamond and anti-PFA axioms to metrization of manifolds, preprint.

Teodor C. Przymusiński, Products of normal spaces, Handbook of set-theoretic topology, North-Holland, Amsterdam, 1984, pp. 781–826. MR 776637

Elliott Pearl (ed.), Open problems in topology. II, Elsevier B. V., Amsterdam, 2007. MR 2367385

Judy Roitman, Basic $S$ and $L$, Handbook of set-theoretic topology, North-Holland, Amsterdam, 1984, pp. 295–326. MR 776626

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

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

Mary Ellen Rudin, A nonmetrizable manifold from $\lozenge ^+$, Topology Appl. 28 (1988), no. 2, 105–112. Special issue on set-theoretic topology. MR 932975, DOI https://doi.org/10.1016/0166-8641%2888%2990002-8

Mary Ellen Rudin and Phillip Zenor, A perfectly normal nonmetrizable manifold, Houston J. Math. 2 (1976), no. 1, 129–134. MR 394560

Saharon Shelah, Proper and improper forcing, 2nd ed., Perspectives in Mathematical Logic, Springer-Verlag, Berlin, 1998. MR 1623206

Chaz Schlindwein, Suslin’s hypothesis does not imply stationary antichains, Ann. Pure Appl. Logic 64 (1993), no. 2, 153–167. MR 1241252, DOI https://doi.org/10.1016/0168-0072%2893%2990032-9

Chaz Schlindwein, SH plus CH does not imply stationary antichains, Ann. Pure Appl. Logic 124 (2003), no. 1-3, 233–265. MR 2013399, DOI https://doi.org/10.1016/S0168-0072%2803%2900057-5

Stevo Todorčević, A dichotomy for P-ideals of countable sets, Fund. Math. 166 (2000), no. 3, 251–267. MR 1809418, DOI https://doi.org/10.4064/fm-166-3-251-267