On compact Hausdorff spaces of countable tightness

Author:
Zoltán T. Balogh

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

Full-text PDF

Abstract | References | Similar Articles | Additional Information

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.

**[A]**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****[A]**-,*There is no "naive" example of a nonseparable sequential bicompact with the Suslin property*, Dokl. Akad. Nauk SSSR**203**(1972), 473-476.**[A]**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****[B]**Zoltán T. Balogh,*Locally nice spaces under Martin’s axiom*, Comment. Math. Univ. Carolin.**24**(1983), no. 1, 63–87. MR**703926****[Ba]**James E. Baumgartner,*Applications of the proper forcing axiom*, Handbook of set-theoretic topology, North-Holland, Amsterdam, 1984, pp. 913–959. MR**776640****[Bu]**Dennis K. Burke,*Closed mappings*, Surveys in general topology, Academic Press, New York-London-Toronto, Ont., 1980, pp. 1–32. MR**564098****[F]**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****[Fr]**D. Fremlin,*Perfect preimages of**and the PFA*, Preprint.**[Fr]**-,*Consequences of Martin's maximum*, Preprint.**[FN]**D. Fremlin and P. Nyikos,*Countably tight, countably compact spaces*, Preprint.**[G]**Gary Gruenhage,*Some results on spaces having an orthobase or a base of subinfinite rank*, Proceedings of the 1977 Topology Conference (Louisiana State Univ., Baton Rouge, La., 1977), I, 1977, pp. 151–159 (1978). MR**540602****[IN]**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**, https://doi.org/10.1016/0166-8641(80)90027-9**[K]**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****[KV]**Kenneth Kunen and Jerry E. Vaughan (eds.),*Handbook of set-theoretic topology*, North-Holland Publishing Co., Amsterdam, 1984. MR**776619****[MM]**R. C. Moore and S. G. Mrowka,*Topologies determined by countable objects*, Notices Amer. Math. Soc.**11**(1964), 554.**[Ny]**Peter Nyikos,*The theory of nonmetrizable manifolds*, Handbook of set-theoretic topology, North-Holland, Amsterdam, 1984, pp. 633–684. MR**776633****[Ny]**-,*Forcing compact non-sequential spaces of countable tightness*, Preprint.**[Ny]**-,*Handwritten manuscript*.**[Ny]**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****[O]**A. J. Ostaszewski,*On countably compact, perfectly normal spaces*, J. London Math. Soc. (2)**14**(1976), no. 3, 505–516. MR**0438292**, https://doi.org/10.1112/jlms/s2-14.3.505**[RZ]**Mary Ellen Rudin and Phillip Zenor,*A perfectly normal nonmetrizable manifold*, Houston J. Math.**2**(1976), no. 1, 129–134. MR**0394560****[Sz]**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****[T]**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**, https://doi.org/10.1090/conm/031/763902**[TP]**Topology Proc.**2**(1977), 679-685.

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC:
03E35,
54A35,
54D30

Retrieve articles in all journals with MSC: 03E35, 54A35, 54D30

Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1989-0930252-6

Keywords:
Proper Forcing Axiom,
countably compact,
compact Hausdorff,
countable tightness,
sequential,
perfect preimage of

Article copyright:
© Copyright 1989
American Mathematical Society