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 ω1 + 1 which is not Fréchet-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 δ + 1 where δ<ω2 with cf(δ) = ω1.

Received 12 June 2017

Research supported by the Spanish Ministry of Education DGI grant MTM2014-59178-P and by the Catalan DURSI grant 2014SGR437.

AMS subject classification: 03E35, 03E04, 03E75, 06E05, 54A25, 54G12

Słowa kluczowe: PCF space, Sequentiality, Fréchet-Urysohn property

[1] 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.

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

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

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

[5] M.R. Burke and M. Magidor, Shelah’s pcf theory and its applications, Annals of Pure and Applied Logic 50:3 (1990), 207–254.

[6] K. Er-Rhaimini and B. Veliˇckovi´c, PCF structures of height less than ω3, The Journal of Symbolic Logic 75:4 (2010), 1231–1248.

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

[8] J.C. Mart´ınez, Cardinal sequences for superatomic Boolean algebras, In: S. Geschke, B. L¨owe and P. Schlicht, editors, Infinity, Computability and Metamathematics, vol. 23 of Tributes Series, pp. 273–284, College Publications, Milton Keynes, 2014.

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

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

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

Czasopismo ukazuje się w sposób ciągły on-line.
Pierwotną formą czasopisma jest wersja elektroniczna.