On PCF spaces which are not Frechet-Urysohn

Juan Carlos Martínez


By means of a forcing argument, it was shown by Pereira that if CH holds then there is a separable PCF space of height w1 + 1 which is not Frechet-Urysohn. In this paper, we give a direct proof of Pereira's theorem by means of a forcing-free argument, and we extend his result to PCF spaces of any height  + 1 where  < w2 with cf() = w1.

Słowa kluczowe: PCF space, Sequentiality, Frechet-Urysohn property.

U. Abraham and M. Magidor, Cardinal arithmetic, In M. Foreman and A. Kanamori, editors, vol. 2 of Handbook of Set Theory, Springer, New York, 2010, pp. 1149-1227.

J. Bagaria, Thin-tall spaces and cardinal sequences, In E. Pearl, editor, Open problems in Topology II, Elsevier, Amsterdam, 2007, pp. 115-124.

J. E. Baumgartner and S. Shelah, Remarks on superatomic Boolean algebras, Annals of Pure and Applied Logic 33:2 (1987), 109-129.

R. Bonnet and M. Rubin, On well-generated Boolean algebras, Annals of Pure and Applied Logic 105:1-3 (2000), 1-50.

M. R. Burke and M. Magidor, Shelah's pcf theory and its applications, Annals of Pure and Applied Logic 50:3 (1990), 207-254.

K. Er-Rhaimini and B. Velickovic, PCF structures of height less than !3, The Journal of Symbolic Logic 75:4 (2010), 1231-1248.

M. Foreman, Some problems in singular cardinals combinatorics, Notre Dame Journal of Formal Logic 46:3 (2005), 309-322.

J. C. Martinez, Cardinal sequences for superatomic Boolean algebras. In S.Geschke, B. Lowe and P. Schlicht, editors, In nity, Computability and Metamathematics, vol. 23 of Tributes Series, pp. 273-284. College Publications, Milton Keynes, 2014.

L. Pereira, Applications of the topological representation of the pcf-structure, Archive for Mathematical Logic 47:5 (2008), 517-527.

J. C. Ruyle, Cardinal sequences of PCF structures, Ph.D. Thesis, University of California, Riverside, 1998.

S. Shelah, Cardinal arithmetic, vol. 29 of Oxford Logic Guides, Oxford University Press, 1994.

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