First countable, countably compact spaces and the continuum hypothesis

Eisworth, Todd; Nyikos, Peter

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

First countable, countably compact spaces and the continuum hypothesis
HTML articles powered by AMS MathViewer

by Todd Eisworth and Peter Nyikos

Trans. Amer. Math. Soc. 357 (2005), 4269-4299

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

Published electronically: June 21, 2005

PDF | Request permission

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 $\omega _2$.

References

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, DOI 10.1090/S0273-0979-1988-15649-2
Zoltán T. Balogh, On compact Hausdorff spaces of countable tightness, Proc. Amer. Math. Soc. 105 (1989), no. 3, 755–764. MR 930252, DOI 10.1090/S0002-9939-1989-0930252-6
Alan Dow, An introduction to applications of elementary submodels to topology, Topology Proc. 13 (1988), no. 1, 17–72. MR 1031969
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
Alan Dow, More set-theory for topologists, Topology Appl. 64 (1995), no. 3, 243–300. MR 1342520, DOI 10.1016/0166-8641(95)00034-E
Todd Eisworth, CH and first countable, countably compact spaces, Topology Appl. 109 (2001), no. 1, 55–73. MR 1804563, DOI 10.1016/S0166-8641(99)00126-1
Todd Eisworth, On perfect pre-images of $\omega _1$, Topology Appl. 125 (2002), no. 2, 263–278. MR 1933576, DOI 10.1016/S0166-8641(01)00280-2
Todd Eisworth, Totally proper forcing and the Moore-Mrówka problem, Fund. Math. 177 (2003), no. 2, 121–137. MR 1992528, DOI 10.4064/fm177-2-2
Todd Eisworth and Judith Roitman, $\textrm {CH}$ with no Ostaszewski spaces, Trans. Amer. Math. Soc. 351 (1999), no. 7, 2675–2693. MR 1638230, DOI 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
D. H. Fremlin, Perfect pre-images of $\omega _i$ and the PFA, Topology Appl. 29 (1988), no. 2, 151–166. MR 949366, DOI 10.1016/0166-8641(88)90072-7
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
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, DOI 10.1016/S0166-8641(01)00140-7
Peter Nyikos, The theory of nonmetrizable manifolds, Handbook of set-theoretic topology, North-Holland, Amsterdam, 1984, pp. 633–684. MR 776633
Saharon Shelah, Nnr revisited, Journal of Symbolic Logic.
Saharon Shelah, Cardinal arithmetic, Oxford Logic Guides, vol. 29, The Clarendon Press, Oxford University Press, New York, 1994. Oxford Science Publications. MR 1318912
—, Jonsson Algebras in an inaccessible $\lambda$ not $\lambda$–Mahlo, Cardinal Arithmetic, Oxford Logic Guides, vol. 29, Oxford University Press, 1994.
—, There are Jonsson algebras in many inaccessible cardinals, Cardinal Arithmetic, Oxford Logic Guides, vol. 29, Oxford University Press, 1994.
Saharon Shelah, Proper and improper forcing, 2nd ed., Perspectives in Mathematical Logic, Springer-Verlag, Berlin, 1998. MR 1623206, DOI 10.1007/978-3-662-12831-2
Jerry E. Vaughan, Countably compact and sequentially compact spaces, Handbook of set-theoretic topology, North-Holland, Amsterdam, 1984, pp. 569–602. MR 776631