Hostname: page-component-7dd5485656-dk7s8 Total loading time: 0 Render date: 2025-10-31T00:18:33.662Z Has data issue: false hasContentIssue false
  • English
  • Français

PFA(S)[S]: More Mutually Consistent Topological Consequences of PFA and V = L

Published online by Cambridge University Press:  20 November 2018

Franklin D. Tall
Franklin D. Tall*
Affiliation:
Department of Mathematics, University of Toronto, Toronto, Ontario M5S 2E4 email: f.tall@utoronto.ca
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Extending the work of Larson and Todorcevic, we show that there is a model of set theory in which normal spaces are collectionwise Hausdorff if they are either first countable or locally compact, and yet there are no first countable $L$ -spaces or compact $S$ -spaces. The model is one of the form $\text{PFA}\left( S \right)\left[ S \right]$ , where $S$ is a coherent Souslin tree.

Keywords

54A3554D1554D2054D4503E3503E5703E65PFA(S)[S]proper forcingcoherent Souslin treelocally compactnormalcollectionwise Hausdorffsupercompact cardinal

Information

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 64 , Issue 5 , 01 October 2012 , pp. 1182 - 1200
Copyright
Copyright © Canadian Mathematical Society 2012

References

[1] Arhangel’skii, A. V., Bicompacta that satisfy the Suslin condition hereditarily. Tightness and free sequences. (Russian) Dokl. Akad. Nauk SSSR 199(1971), 1227–1230.Google Scholar
[2] Arhangel’skii, A. V., The property of paracompactness in the class of perfectly normal locally bicompact spaces. (Russian) Dokl. Akad. Nauk SSSR 203(1972), 1231–1234.Google Scholar
[3] Baumgartner, J. E., Taylor, A. D., and S.Wagon, Structural properties of ideals. Dissertationes Math. 197(1982), 1–95.Google Scholar
[4] Devlin, K. J., The Yorkshireman's guide to proper forcing. In: Surveys in set theory, London Math. Soc. Lecture Note Ser., 87, Cambridge University Press, Cambridge, 1983, pp. 60–115.Google Scholar
[5] Dow, A., On the consistency of the Moore-Mrowka solution. Proceedings of the Symposium on General Topology and Applications (Oxford, 1989). Topology Appl. 44(1992), no. 1–3, 125–141. http://dx.doi.org/10.1016/0166-8641(92)90085-E Google Scholar
[6] Farah, I., OCA and towers in P(N)/Fin. Comment. Math. Univ. Carolin. 37(1996), no. 4, 861–866.Google Scholar
[7] Fleissner, W. G., Normal Moore spaces in the constructible universe. Proc. Amer. Math. Soc. 46(1974), 294–298. http://dx.doi.org/10.1090/S0002-9939-1974-0362240-4 Google Scholar
[8] Gruenhage, G. and Koszmider, P., The Arkhangel’skiı-Tall problem under Martin's axiom. Fund. Math. 149(1996), no. 3, 275–285.Google Scholar
[9] Jech, T., Set theory. The third millenium edition, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2003.Google Scholar
[10] Kunen, K. and Tall, F. D., Between Martin's axiom and Souslin's hypothesis. Fund. Math. 102(1979), no. 3, 173–181.Google Scholar
[11] König, B., Local coherence. Ann. Pure Appl. Logic 124(2003), no. 1–3, 107–139. http://dx.doi.org/10.1016/S0168-0072(03)00053-8 Google Scholar
[12] Larson, P., An Smax variation for one Souslin tree. J. Symbolic Logic 64(1999), no. 1, 81–98. http://dx.doi.org/10.2307/2586753 Google Scholar
[13] Larson, P. and Tall, F. D., Locally compact perfectly normal spaces may all be paracompact. Fund. Math. 210(2010), no. 3, 285–300. http://dx.doi.org/10.4064/fm210-3-4 Google Scholar
[14] Larson, P. and Tall, F. D., On the hereditary paracompactness of locally compact hereditarily normal spaces. Canad. Math. Bull., to appear.Google Scholar
[15] Larson, P. and Todorcevic, S., Katětov's problem. Trans. Amer. Math. Soc. 354(2002), no. 5, 1783–1791. http://dx.doi.org/10.1090/S0002-9947-01-02936-1 Google Scholar
[16] Laver, R., Making the supercompactness of indestructible under C-directed closed forcing. Israel J. Math. 29(1978), no. 4, 385–388. http://dx.doi.org/10.1007/BF02761175 Google Scholar
[17] Miyamoto, T., !1-Souslin trees under countable support iterations. Fund. Math. 142(1993), no. 3, 257–261.Google Scholar
[18] Tall, F. D., Normality versus collectionwise normality. In: Handbook of set-theoretic topology, North-Holland, Amsterdam, 1984, pp. 685–732.Google Scholar
[19] Tall, F. D., PFA(S)[S] and locally compact normal spaces. Topology Appl., to appear.Google Scholar
[20] Tall, F. D., Problems arising from Z. T. Balogh's “Locally nice spaces under Martin's axiom” [Comment. Math. Univ. Carolin. 24(1983), no. 1, 63–87]. Topology Appl. 151(2005), no. 1–3, 215–225. http://dx.doi.org/10.1016/j.topol.2004.07.014 Google Scholar
[21] Tall, F. D., Covering and separation properties in the Easton model. Topology Appl. 28(1988), no. 2, 155–163. http://dx.doi.org/10.1016/0166-8641(88)90007-7 Google Scholar
[22] Tall, F. D., Set-theoretic consistency results and topological theorems concerning the normal Moore space conjecture and related problems. Dissertationes Math. (Rozprawy Mat.) 148(1977), 1–53.Google Scholar
[23] Tall, F. D., PFA(S)[S] and the Arhangel’skiı-Tall problem. Topology Proc. 40(2012), 99–108.Google Scholar
[24] Taylor, A. D., Diamond principles, ideals and the normal Moore space problem. Canad. J. Math. 33(1981), no. 2, 282–296. http://dx.doi.org/10.4153/CJM-1981-023-4 Google Scholar
[25] Todorcevic, S., Directed sets and cofinal types. Trans. Amer. Math. Soc. 290(1985), no. 2, 711–723. http://dx.doi.org/10.2307/2000309 Google Scholar
[26] Watson, W. S., Sixty questions on regular not paracompact spaces. Proceedings of the 11th Winter School on Abstract Analysis (Železná Ruda, 1983). Rend. Circ. Mat. Palermo (2) 1984, Suppl. 3, 369–373.Google Scholar
[27] Watson, W. S., Locally compact normal spaces in the constructible universe. Canad. J. Math. 34(1982), no. 5, 1091–1096. http://dx.doi.org/10.4153/CJM-1982-078-8Google Scholar