First countable, countably compact spaces and the continuum hypothesis

Eisworth, Todd; Nyikos, Peter

doi:10.1090/S0002-9947-05-04034-1

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First countable, countably compact spaces and the continuum hypothesis

Authors: Todd Eisworth and Peter Nyikos
Journal: Trans. Amer. Math. Soc. 357 (2005), 4269-4299
MSC (2000): Primary 03E75
Posted: June 21, 2005
MathSciNet review: 2156711
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Abstract | References | Similar Articles | Additional Information

Abstract: We build a model of ZFC+CH in which every first countable, countably compact space is either compact or contains a homeomorphic copy of $\omega_1$ with the order topology. The majority of the paper consists of developing forcing technology that allows us to conclude that our iteration adds no reals. Our results generalize Saharon Shelah's iteration theorems appearing in Chapters V and VIII of Proper and improper forcing (1998), as well as Eisworth and Roitman's (1999) iteration theorem. We close the paper with a ZFC example (constructed using Shelah's club-guessing sequences) that shows similar results do not hold for closed pre-images of $\omega_2$ .

1. Z. Balogh, A. Dow, D. H. Fremlin, and P. J. Nyikos, Countable tightness and proper forcing, Bull. Amer. Math. Soc. (N.S.) 19 (1988), no. 1, 295–298. MR 940491 (89e:03088), http://dx.doi.org/10.1090/S0273-0979-1988-15649-2
2. Zoltán T. Balogh, On compact Hausdorff spaces of countable tightness, Proc. Amer. Math. Soc. 105 (1989), no. 3, 755–764. MR 930252 (89h:03088), http://dx.doi.org/10.1090/S0002-9939-1989-0930252-6
3. Alan Dow, An introduction to applications of elementary submodels to topology, Topology Proc. 13 (1988), no. 1, 17–72. MR 1031969 (91a:54003)
4. Miroslav Hušek and Jan van Mill (eds.), Recent progress in general topology, North-Holland Publishing Co., Amsterdam, 1992. Papers from the Symposium on Topology (Toposym) held in Prague, August 19–23, 1991. MR 1229121 (95g:54004)
5. Alan Dow, More set-theory for topologists, Topology Appl. 64 (1995), no. 3, 243–300. MR 1342520 (97a:54005), http://dx.doi.org/10.1016/0166-8641(95)00034-E
6. Todd Eisworth, CH and first countable, countably compact spaces, Topology Appl. 109 (2001), no. 1, 55–73. MR 1804563 (2001k:54010), http://dx.doi.org/10.1016/S0166-8641(99)00126-1
7. Todd Eisworth, On perfect pre-images of 𝜔₁, Topology Appl. 125 (2002), no. 2, 263–278. MR 1933576 (2003g:54005), http://dx.doi.org/10.1016/S0166-8641(01)00280-2
8. Todd Eisworth, Totally proper forcing and the Moore-Mrówka problem, Fund. Math. 177 (2003), no. 2, 121–137. MR 1992528 (2004g:03093), http://dx.doi.org/10.4064/fm177-2-2
9. Todd Eisworth and Judith Roitman, 𝐶𝐻 with no Ostaszewski spaces, Trans. Amer. Math. Soc. 351 (1999), no. 7, 2675–2693. MR 1638230 (2000b:03182), http://dx.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 (91c:54001)
11. D. H. Fremlin, Perfect pre-images of 𝜔ᵢ and the PFA, Topology Appl. 29 (1988), no. 2, 151–166. MR 949366 (89i:03096), http://dx.doi.org/10.1016/0166-8641(88)90072-7
12. Martin Goldstern, Tools for your forcing construction, Set theory of the reals (Ramat Gan, 1991) Israel Math. Conf. Proc., vol. 6, Bar-Ilan Univ., Ramat Gan, 1993, pp. 305–360. MR 1234283 (94h:03102)
13. Gary Gruenhage, Spaces having a small diagonal, Proceedings of the International Conference on Topology and its Applications (Yokohama, 1999), 2002, pp. 183–200. MR 1919300 (2003g:54050), http://dx.doi.org/10.1016/S0166-8641(01)00140-7
14. Peter Nyikos, The theory of nonmetrizable manifolds, Handbook of set-theoretic topology, North-Holland, Amsterdam, 1984, pp. 633–684. MR 776633 (86f:54054)
15. Saharon Shelah, Nnr revisited, Journal of Symbolic Logic.
16. Saharon Shelah, Cardinal arithmetic, Oxford Logic Guides, vol. 29, The Clarendon Press Oxford University Press, New York, 1994. Oxford Science Publications. MR 1318912 (96e:03001)
17. -, Jonsson Algebras in an inaccessible $\lambda$ not $\lambda$ -Mahlo, Cardinal Arithmetic, Oxford Logic Guides, vol. 29, Oxford University Press, 1994.
18. -, There are Jonsson algebras in many inaccessible cardinals, Cardinal Arithmetic, Oxford Logic Guides, vol. 29, Oxford University Press, 1994.
19. Saharon Shelah, Proper and improper forcing, 2nd ed., Perspectives in Mathematical Logic, Springer-Verlag, Berlin, 1998. MR 1623206 (98m:03002)
20. Jerry E. Vaughan, Countably compact and sequentially compact spaces, Handbook of set-theoretic topology, North-Holland, Amsterdam, 1984, pp. 569–602. MR 776631 (86c:54022)

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Additional Information

Todd Eisworth
Affiliation: Department of Mathematics, University of Northern Iowa, Cedar Falls, Iowa 50613
Address at time of publication: Department of Mathematics, Ohio University, Athens, Ohio 45701
Email: eisworth@uni.edu, eisworth@math.ohiou.edu

Peter Nyikos
Affiliation: Department of Mathematics, University of South Carolina, Columbia, South Carolina 29208
Email: nyikos@math.sc.edu

DOI: http://dx.doi.org/10.1090/S0002-9947-05-04034-1
PII: S 0002-9947(05)04034-1
Keywords: Proper forcing, iterations, Continuum Hypothesis, pre--images of $\omega_1$
Received by editor(s): May 23, 2002
Posted: June 21, 2005
Additional Notes: The first author was partially supported by a Summer Fellowship granted by the Graduate College of the University of Northern Iowa
Article copyright: © Copyright 2005 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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