Transitivity and the 𝛾-space conjecture in ordered spaces

Kofner, Jacob

doi:10.1090/S0002-9939-1981-0601744-7

Transitivity and the $\gamma$ -space conjecture in ordered spaces

Author: Jacob Kofner
Journal: Proc. Amer. Math. Soc. 81 (1981), 629-635
MSC: Primary 54F05; Secondary 54E15
DOI: https://doi.org/10.1090/S0002-9939-1981-0601744-7
MathSciNet review: 601744
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Abstract: Each generalized ordered $\gamma$ -space is nonarchimedean quasimetrizable. Moreover, each generalized ordered space is -transitive, i.e. for each neighbournet there is a transitive neighbournet $V \subset {U^3}$ .

[B] H. R. Bennett, Quasimetrizability and the 𝛾-space property in certain generalized ordered spaces, The Proceedings of the 1979 Topology Conference (Ohio Univ., Athens, Ohio, 1979), 1979, pp. 1–12 (1980). MR 583684
[ELu] R. Engelking and D. Lutzer, Paracompactness in ordered spaces, Fund. Math. 94 (1977), no. 1, 49–58. MR 0428278, https://doi.org/10.4064/fm-94-1-25-33
[FL1] P. Fletcher and W. F. Lindgren, Transitive quasi-uniformities, J. Math. Anal. Appl. 39 (1972), 397–405. MR 0317281, https://doi.org/10.1016/0022-247X(72)90210-7
[FL2] Peter Fletcher and William F. Lindgren, Quasi-uniformities with a transitive base, Pacific J. Math. 43 (1972), 619–631. MR 0319155
[FL3] -, Quasiuniform spaces (to appear).
[F] R. Fox, On metrizability and quasi-metrizability (to appear).
[G] Gary Gruenhage, A note on quasi-metrizability, Canad. J. Math. 29 (1977), no. 2, 360–366. MR 0436089, https://doi.org/10.4153/CJM-1977-039-2
[H] R. E. Hodel, Spaces defined by sequences of open covers which guarantee that certain sequences have cluster points, Duke Math. J. 39 (1972), 253–263. MR 0293580
[J1] H. Junnila, Covering properties and quasi-uniformities of topological spaces, Ph.D. thesis, Virginia Polytech. Inst. and State Univ., 1978.
[J2] Heikki J. K. Junnila, Neighbornets, Pacific J. Math. 76 (1978), no. 1, 83–108. MR 0482677
[K] Ja. A. Kofner, Δ-metrizable spaces, Mat. Zametki 13 (1973), 277–287 (Russian). MR 0324664
[LF] W. F. Lindgren and P. Fletcher, Locally quasi-uniform spaces with countable bases, Duke Math. J. 41 (1974), 231–240. MR 0341422
[Lu] D. J. Lutzer, On generalized ordered spaces, Dissertationes Math. Rozprawy Mat. 89 (1971), 32. MR 0324668
[N1] V. W. Niemytzki, On the “third axiom of metric space”, Trans. Amer. Math. Soc. 29 (1927), no. 3, 507–513. MR 1501402, https://doi.org/10.1090/S0002-9947-1927-1501402-2
[N2] -, Über die Axioms des metrischen Raumes, Math. Ann. 106 (1931), 666-671.
[T] Topology Proceedings 2 (1977), p. 687.
[W] W. A. Wilson, On Quasi-Metric Spaces, Amer. J. Math. 53 (1931), no. 3, 675–684. MR 1506845, https://doi.org/10.2307/2371174