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Cosmicity of cometrizable spaces


Author: Gary Gruenhage
Journal: Trans. Amer. Math. Soc. 313 (1989), 301-315
MSC: Primary 54E20; Secondary 54A35
DOI: https://doi.org/10.1090/S0002-9947-1989-0992600-5
MathSciNet review: 992600
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Abstract: A space $ X$ is cometrizable if $ X$ has a coarser metric topology such that each point of $ X$ has a neighborhood base of metric closed sets. Most examples in the literature of spaces obtained by modifying the topology of the plane or some other metric space are cometrizable. Assuming the Proper Forcing Axiom (PFA) we show that the following statements are equivalent for a cometrizable space $ X$ : (a) $ X$ is the continuous image of a separable metric space; (b) $ {X^\omega }$ is hereditarily separable and hereditarily Lindelöf, (c) $ {X^2}$ has no uncountable discrete subspaces; (d) $ X$ is a Lindelöf semimetric space; (e) $ X$ has the pointed $ {\text{ccc}}$. This result is a corollary to our main result which states that, under PFA, if $ X$ is a cometrizable space with no uncountable discrete subspaces, then either $ X$ is the continuous image of a separable metric space or $ X$ contains a copy of an uncountable subspace of the Sorgenfrey line.


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  • [A] A. V. Arhangel' skii, The structure and classification of topological spaces and cardinal invariants, Russian Math. Surveys 33 (1978), 33-96. MR 526012 (80i:54005)
  • [ARS] U. Abraham, M. Rubin, and S. Shelah, On the consistency of some partition theorems for continuous colorings, and the structure of $ {\aleph _1}$-dense real order types, Ann. Pure Appl. Logic 29 (1985), 123-206. MR 801036 (87d:03132)
  • [AS] U. Abraham and S. Shelah, Martin's Axiom does not imply that every two $ {\aleph _1}$-dense sets of reals are isomorphic, Israel J. Math. 38 (1981), 161-176. MR 599485 (82a:03048)
  • [B$ _{1}$] J. Baumgartner, Applications of the proper forcing axiom, Handbook of Set-Theoretic Topology, edited by K. Kunen and J. E. Vaughan, North-Holland, Amsterdam, 1984, pp. 914-959. MR 776640 (86g:03084)
  • [B$ _{2}$] -, All $ {\aleph _1}$-dense sets of reals can be isomorphic, Fund. Math. 79 (1973), 101-106. MR 0317934 (47:6483)
  • [Br] E. S. Berney, A regular Lindelöf semi-metric space which has no countable network, Proc. Amer. Math. Soc. 26 (1970), 361-364. MR 0270336 (42:5225)
  • [Bs] A. Beslagic, Embedding cosmic spaces in Lusin spaces, Proc. Amer. Math. Soc. 89 (1983), 515-518. MR 715877 (84h:54037)
  • [BvD] D. K. Burke and E. K. van Douwen, No dense metrizable $ {G_\delta }$-subspaces in butterfly semimetrizable spaces, Topology Appl. 11 (1980), 31-36. MR 550870 (83a:54030)
  • [C] K. Ciesielski, Martin's Axiom and a regular topological space with uncountable netweight whose countable product is hereditarily separable and hereditarily Lindelöf, J. Symbolic Logic 52 (1987), 396-399. MR 890447 (89d:03047)
  • [vDK] E. van Douwen and K. Kunen, $ L$-spaces and $ S$-spaces in $ P(\omega)$, Topology Appl. 14 (1982), 143-149. MR 667660 (83k:54003)
  • [Fr$ _{1}$] D. H. Fremlin, Consequences of Martin's Axiom, Cambridge Univ. Press, 1984.
  • [Fr$ _{2}$] -, Notes on Martin's maximum, unpublished notes.
  • [G] G. Gruenhage, On the existence of metrizable or Sorgenfrey subspaces, General Topology and its Relation to Modern Algebra and Analysis (Proc. Sixth Prague Topology Sympos., 1986, (Z. Frolik, ed.), Heldermann-Verlag, Berlin, 1986, pp. 223-230. MR 952608 (89g:54013)
  • [H] R. W. Heath, On certain first-countable spaces, Topology Seminar (Wis., 1965), Ann. of Math. Studies, no 60, Princeton Univ. Press, Princeton, N.J., 1966, pp. 103-113.
  • [J] I. Juhasz, SETOP Conference notes, Toronto, Canada, 1980.
  • [JKR] I. Juhasz, K. Kunen, and M. E. Rudin, Two more hereditarily separable non-Lindelöf spaces, Canad. J. Math. 28 (1976), 998-1005. MR 0428245 (55:1270)
  • [M] E. A. Michael, Paracompactness and the Lindelöf property in finite and countable Cartesian products, Comput. Math. 23 (1971), 199-214. MR 0287502 (44:4706)
  • [S] S. Shelah, Proper forcing, Lecture Notes in Math., vol. 940, Springer-Verlag, Berlin, 1982. MR 675955 (84h:03002)
  • [Sz] Z. Szentmiklossy, $ S$-spaces and $ L$-spaces under Martin's Axiom, Colloq. Math. Soc. János Bolyai 23 (Budapest 1978 (II)), North-Holland, Amsterdam, 1980, pp. 1139-1145. MR 588860 (81k:54032)
  • [Ta] F. Tall, The density topology, Pacific J. Math. 75 (1976), 275-284. MR 0419709 (54:7727)
  • [Tk] M. G. Tkačenko, Chains and cardinals, Dokl. Akad. Nauk SSSR 239 (1978), 546-549. MR 0500798 (58:18329)
  • [To$ _{1}$] S. Todorcevic, Forcing positive partition relations, Trans. Amer. Math. Soc. 280 (1983), 703-720. MR 716846 (85d:03102)
  • [To$ _{2}$] -, Remarks on cellularity in products, Compositio Math. 57 (1986), 357-372. MR 829326 (88h:54009)
  • [To$ _{3}$] -, A class of spaces associated with gaps, unpublished manuscript.
  • [V] N. V. Veličko, Symmetrizable spaces, Mat. Zametki 12 (1972), 577-582=Math. Notes 12 (1972), 784-786. MR 0326678 (48:5021)

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DOI: https://doi.org/10.1090/S0002-9947-1989-0992600-5
Article copyright: © Copyright 1989 American Mathematical Society

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