On compact Hausdorff spaces of countable tightness
HTML articles powered by AMS MathViewer
- by Zoltán T. Balogh PDF
- Proc. Amer. Math. Soc. 105 (1989), 755-764 Request permission
Abstract:
A general combinatorial theorem for countably compact, noncompact spaces is given under the Proper Forcing Axiom. It follows that compact Hausdorff spaces of countable tightness are sequential under PFA, solving the Moore-Mrowka Problem. Other applications are also given.References
- A. V. Arhangelskij, A survey of some recent advances in general topology, old and new problems, Actes du Congrès International des Mathématiciens (Nice, 1970) Gauthier-Villars, Paris, 1971, pp. 19–26. MR 0428244 —, There is no "naive" example of a nonseparable sequential bicompact with the Suslin property, Dokl. Akad. Nauk SSSR 203 (1972), 473-476.
- A. V. Arhangel′skiĭ, The structure and classification of topological spaces and cardinal invariants, Uspekhi Mat. Nauk 33 (1978), no. 6(204), 29–84, 272 (Russian). MR 526012
- Zoltán T. Balogh, Locally nice spaces under Martin’s axiom, Comment. Math. Univ. Carolin. 24 (1983), no. 1, 63–87. MR 703926
- James E. Baumgartner, Applications of the proper forcing axiom, Handbook of set-theoretic topology, North-Holland, Amsterdam, 1984, pp. 913–959. MR 776640
- Dennis K. Burke, Closed mappings, Surveys in general topology, Academic Press, New York-London-Toronto, Ont., 1980, pp. 1–32. MR 564098
- V. V. Fedorčuk, Completely closed mappings, and the compatibility of certain general topology theorems with the axioms of set theory, Mat. Sb. (N.S.) 99 (141) (1976), no. 1, 3–33, 135 (Russian). MR 0410631 D. Fremlin, Perfect preimages of ${\omega _1}$ and the PFA, Preprint. —, Consequences of Martin’s maximum, Preprint. D. Fremlin and P. Nyikos, Countably tight, countably compact spaces, Preprint.
- Gary Gruenhage, Some results on spaces having an orthobase or a base of subinfinite rank, Topology Proc. 2 (1977), no. 1, 151–159 (1978). MR 540602
- Mohammad Ismail and Peter Nyikos, On spaces in which countably compact sets are closed, and hereditary properties, Topology Appl. 11 (1980), no. 3, 281–292. MR 585273, DOI 10.1016/0166-8641(80)90027-9
- Kenneth Kunen, Set theory, Studies in Logic and the Foundations of Mathematics, vol. 102, North-Holland Publishing Co., Amsterdam-New York, 1980. An introduction to independence proofs. MR 597342
- Kenneth Kunen and Jerry E. Vaughan (eds.), Handbook of set-theoretic topology, North-Holland Publishing Co., Amsterdam, 1984. MR 776619 R. C. Moore and S. G. Mrowka, Topologies determined by countable objects, Notices Amer. Math. Soc. 11 (1964), 554.
- Peter Nyikos, The theory of nonmetrizable manifolds, Handbook of set-theoretic topology, North-Holland, Amsterdam, 1984, pp. 633–684. MR 776633 —, Forcing compact non-sequential spaces of countable tightness, Preprint. —, Handwritten manuscript.
- P. Nyikos, Progress on countably compact spaces, General topology and its relations to modern analysis and algebra, VI (Prague, 1986) Res. Exp. Math., vol. 16, Heldermann, Berlin, 1988, pp. 379–410. MR 952624
- A. J. Ostaszewski, On countably compact, perfectly normal spaces, J. London Math. Soc. (2) 14 (1976), no. 3, 505–516. MR 438292, DOI 10.1112/jlms/s2-14.3.505
- Mary Ellen Rudin and Phillip Zenor, A perfectly normal nonmetrizable manifold, Houston J. Math. 2 (1976), no. 1, 129–134. MR 394560
- Z. Szentmiklóssy, $S$-spaces and $L$-spaces under Martin’s axiom, Topology, Vol. II (Proc. Fourth Colloq., Budapest, 1978) Colloq. Math. Soc. János Bolyai, vol. 23, North-Holland, Amsterdam-New York, 1980, pp. 1139–1145. MR 588860
- Stevo Todorčević, A note on the proper forcing axiom, Axiomatic set theory (Boulder, Colo., 1983) Contemp. Math., vol. 31, Amer. Math. Soc., Providence, RI, 1984, pp. 209–218. MR 763902, DOI 10.1090/conm/031/763902 Topology Proc. 2 (1977), 679-685.
Additional Information
- © Copyright 1989 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 105 (1989), 755-764
- MSC: Primary 03E35; Secondary 54A35, 54D30
- DOI: https://doi.org/10.1090/S0002-9939-1989-0930252-6
- MathSciNet review: 930252