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Analysis of a topological basis problem


We examine a basis problem for uncountable regular first countable spaces using the Proper Forcing Axiom. We introduce a notion of inner and outer topologies and show that they come quite close to characterizing the correctness of the current conjecture about this basis problem.

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  1. J. E. Baumgartner, All \(\aleph_1\)-dense sets of reals can be isomorphic, Fund. Math., 79 (1973), 101–106.

  2. E. K. van Douwen and W. F. Pfeffer, Some properties of the Sorgenfrey line and related spaces, Pacific J. Math., 81 (1979), 371–377.

  3. R. Engelking, Topology, 2nd ed., Sigma Series in Pure Mathematics, vol. 6, Heldermann Verlag (Berlin, 1989).

  4. D. H. Fremlin, Consequences of Martin’s Axiom, Cambridge Univ. Press (1984).

  5. G. Gruenhage, On the existence of metrizable or Sorgenfrey subspaces, in: General Topology and its Relations to Modern Analysis and Algebra, VI (Prague, 1986), Heldermann Verlag (Berlin, 1988), pp. 223–230.

  6. G. Gruenhage, Cosmicity of cometrizable spaces, Trans. Amer. Math. Soc., 313 (1989), 301–315.

  7. T. Jech, Set Theory, The third millennium edition, revised and expanded, Springer Monographs in Mathematics, Springer-Verlag (Berlin, 2003).

  8. L. Kronecker, Näherungsweise ganzzahlige Auflösung linearer Gleichungen, Monatsber. Königl. Preuss. Akad. Wiss. Berlin (1884), 1179–1193, 1271–1299.

  9. J. T. Moore, A solution to the L space problem, J. Amer. Math. Soc., 19 (2006) 717–736.

  10. J.T. Moore, An L space with a d-separable square, Topology Appl., 155 (2008), 304–307.

  11. Y. Peng, An L space with non-Lindelöf square, Topology Proc., 46 (2015), 233–242.

  12. Y. Peng and L. Wu, A Lindelöf group with non-Lindelöf square, Adv. in Math., 325 (2018), 215–252.

  13. W. Sierpiński, Sur un problѐme de la théorie des relations, Ann. Scuola Norm. Sup.Pisa CI. Sci. (2), 2 (1933), 285–287.

  14. S. Todorcevic, Forcing positive partition relations, Trans. Amer. Math. Soc., 280(1983), 703–720.

  15. S. Todorcevic, Partitioning pairs of countable ordinals, Acta Math., 159 (1987), 261–294.

  16. S. Todorcevic, Partition Problems in Topology, Amer. Math. Soc. (Providence, RI,1989).

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Correspondence to Y. Peng.

Additional information

The research of the second author was partially supported by grants from NSERC (455916) and CNRS (UMR7586).

Peng was partially supported by NSFC No. 11901562 and a program of the Chinese Academy of Sciences.

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Peng, Y., Todorcevic, S. Analysis of a topological basis problem. Acta Math. Hungar. 167, 419–475 (2022).

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Key words and phrases

  • basis problem
  • countable network
  • Sorgenfrey space

Mathematics Subject Classification

  • 54A35
  • 54D20
  • 03E75