Martin’s Axiom does not imply perfectly normal non-archimedean spaces are metrizable

Qiao, Yuan-Qing

doi:10.1090/S0002-9939-00-04940-6

Martin’s Axiom does not imply perfectly normal non-archimedean spaces are metrizable
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by Yuan-Qing Qiao PDF

Proc. Amer. Math. Soc. 129 (2001), 1179-1188 Request permission

Abstract:

In this paper we prove that in various models of Martin’s Axiom there are perfectly normal, non-metrizable non-archimedean spaces of $\aleph _2$.

References

James E. Baumgartner, Iterated forcing, Surveys in set theory, London Math. Soc. Lecture Note Ser., vol. 87, Cambridge Univ. Press, Cambridge, 1983, pp. 1–59. MR 823775, DOI 10.1017/CBO9780511758867.002
Keith J. Devlin, Constructibility, Perspectives in Mathematical Logic, Springer-Verlag, Berlin, 1984. MR 750828, DOI 10.1007/978-3-662-21723-8
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
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
Peter Nyikos, A survey of zero-dimensional spaces, Topology (Proc. Ninth Annual Spring Topology Conf., Memphis State Univ., Memphis, Tenn., 1975) Lecture Notes in Pure and Appl. Math., Vol. 24, Dekker, New York, 1976, pp. 87–114. MR 0442870
P. Nyikos and H. C. Reichel, On the structure of zerodimensional spaces, Nederl. Akad. Wetensch. Proc. Ser. A 78=Indag. Math. 37 (1975), 120–136. MR 0365527, DOI 10.1016/1385-7258(75)90024-4
S. Purisch, The orderability of non-Archimedean spaces, Topology Appl. 16 (1983), no. 3, 273–277. MR 722120, DOI 10.1016/0166-8641(83)90024-X
Y. Q. Qiao, On Non-archimedean Spaces, Thesis, Univ. of Toronto, 1992.
Yuan-Qing Qiao and Franklin D. Tall, Perfectly normal non-Archimedean spaces in Mitchell models, Topology Proc. 18 (1993), 231–244. MR 1305135
Y. Q. Qiao and F. D. Tall, Perfectly normal non-metrizable non-archimedean spaces are generalized Souslin lines, Proc. Amer. Math. Soc. (to appear).
Saharon Shelah, Proper forcing, Lecture Notes in Mathematics, vol. 940, Springer-Verlag, Berlin-New York, 1982. MR 675955, DOI 10.1007/978-3-662-21543-2
Saharon Shelah and Stevo Todorčević, A note on small Baire spaces, Canad. J. Math. 38 (1986), no. 3, 659–665. MR 845670, DOI 10.4153/CJM-1986-033-8
Stevo B. Todorčević, Some consequences of $\textrm {MA}+\neg \textrm {wKH}$, Topology Appl. 12 (1981), no. 2, 187–202. MR 612015, DOI 10.1016/0166-8641(81)90020-1
Stevo B. Todorčević, Trees, subtrees and order types, Ann. Math. Logic 20 (1981), no. 3, 233–268. MR 631563, DOI 10.1016/0003-4843(81)90005-X