Solution of the 𝛾-space problem

Solution of the $\gamma$-space problem
HTML articles powered by AMS MathViewer

by Ralph Fox

Proc. Amer. Math. Soc. 85 (1982), 606-608

DOI: https://doi.org/10.1090/S0002-9939-1982-0660614-X

PDF | Request permission

This paper disproves the classic conjecture that every $\gamma$-space is quasi-metrizable.

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
Peter Fletcher and William F. Lindgren, Quasi-uniform spaces, Lecture Notes in Pure and Applied Mathematics, vol. 77, Marcel Dekker, Inc., New York, 1982. MR 660063
Ralph Fox, Pretransitivity and products of suborderable spaces, Topology and order structures, Part 1 (Lubbock, Tex., 1980) Math. Centre Tracts, vol. 142, Math. Centrum, Amsterdam, 1981, pp. 115–118. MR 630545

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
Heikki J. K. Junnila, Neighbornets, Pacific J. Math. 76 (1978), no. 1, 83–108. MR 482677, DOI 10.2140/pjm.1978.76.83

Covering properties and quasi-uniformities of topological spaces

Jacob Kofner, Transitivity and the $\gamma$-space conjecture in ordered spaces, Proc. Amer. Math. Soc. 81 (1981), no. 4, 629–635. MR 601744, DOI 10.1090/S0002-9939-1981-0601744-7

Transitivity and orthobases

W. F. Lindgren and P. Fletcher, Locally quasi-uniform spaces with countable bases, Duke Math. J. 41 (1974), 231–240. MR 341422, DOI 10.1215/S0012-7094-74-04125-8
W. F. Lindgren and P. J. Nyikos, Spaces with bases satisfying certain order and intersection properties, Pacific J. Math. 66 (1976), no. 2, 455–476. MR 445452, DOI 10.2140/pjm.1976.66.455
S. Ĭ. Nedev and M. M. Čoban, On the theory of $o$-metrizable spaces. III, Vestnik Moskov. Univ. Ser. I Mat. Meh. 27 (1972), no. 3, 10–15 (Russian, with English summary). MR 0307192
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
Hugo Ribeiro, Sur les espaces à métrique faible, Portugal. Math. 4 (1943), 21–40. MR 9432
Hugo Ribeiro, Corrections à la note “Sur les espaces à métrique faible” (Port. Math., Vol. 4, (1943) Fasc. 1, p. 21–40), Portugal. Math. 4 (1943), 65–68 (French). MR 10266
Ralph R. Sabella, Convergence properties of neighboring sequences, Proc. Amer. Math. Soc. 38 (1973), 405–409. MR 312479, DOI 10.1090/S0002-9939-1973-0312479-8

2

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