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
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Proc. Amer. Math. Soc. 81 (1981), 629-635 Request permission

Abstract:

Each generalized ordered $\gamma$-space is nonarchimedean quasimetrizable. Moreover, each generalized ordered space is $3$-transitive, i.e. for each neighbournet $U$ there is a transitive neighbournet $V \subset {U^3}$.

References

H. R. Bennett, Quasimetrizability and the $\gamma$-space property in certain generalized ordered spaces, Topology Proc. 4 (1979), no. 1, 1–12 (1980). MR 583684
R. Engelking and D. Lutzer, Paracompactness in ordered spaces, Fund. Math. 94 (1977), no. 1, 49–58. MR 428278, DOI 10.4064/fm-94-1-25-33
P. Fletcher and W. F. Lindgren, Transitive quasi-uniformities, J. Math. Anal. Appl. 39 (1972), 397–405. MR 317281, DOI 10.1016/0022-247X(72)90210-7
Peter Fletcher and William F. Lindgren, Quasi-uniformities with a transitive base, Pacific J. Math. 43 (1972), 619–631. MR 319155

Quasiuniform spaces

On metrizability and quasi-metrizability

Gary Gruenhage, A note on quasi-metrizability, Canadian J. Math. 29 (1977), no. 2, 360–366. MR 436089, DOI 10.4153/CJM-1977-039-2
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 293580

Covering properties and quasi-uniformities of topological spaces

Heikki J. K. Junnila, Neighbornets, Pacific J. Math. 76 (1978), no. 1, 83–108. MR 482677
Ja. A. Kofner, $\Delta$-metrizable spaces, Mat. Zametki 13 (1973), 277–287 (Russian). MR 324664
W. F. Lindgren and P. Fletcher, Locally quasi-uniform spaces with countable bases, Duke Math. J. 41 (1974), 231–240. MR 341422
D. J. Lutzer, On generalized ordered spaces, Dissertationes Math. (Rozprawy Mat.) 89 (1971), 32. MR 324668
V. W. Niemytzki, On the “third axiom of metric space”, Trans. Amer. Math. Soc. 29 (1927), no. 3, 507–513. MR 1501402, DOI 10.1090/S0002-9947-1927-1501402-2

Über die Axioms des metrischen Raumes

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W. A. Wilson, On Quasi-Metric Spaces, Amer. J. Math. 53 (1931), no. 3, 675–684. MR 1506845, DOI 10.2307/2371174