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Another view of metrizability


Author: H. H. Hung
Journal: Proc. Amer. Math. Soc. 101 (1987), 551-554
MSC: Primary 54E35
DOI: https://doi.org/10.1090/S0002-9939-1987-0908667-X
MathSciNet review: 908667
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Abstract: A fact long considered unsatisfactory about the classical metrization theorem of Alexandroff-Urysohn is that it expresses metrizability as a countable uniformity, uniformity itself being almost the former. In view of their unification, the classical theorems, with the exception of Arhangel'skiĭ's regular open base theorem, are all really subject to the same criticism, to which our theorem here is an answer. We give a generalization here of Arhangel'skiĭ's, of which Arhangel'skiĭ's itself, the fundamental theorem of Alexandroff-Urysohn, A. H. Frink's, and the Double Sequence Theorem of Nagata are all obvious special cases.


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  • [1] P. S. Alexandroff and P. Urysohn, Une condition necessaire et suffisante pour qu'une class $ (\mathcal{L})$ soit une class $ (\mathcal{D})$, C. R. Acad. Sci. Paris 177 (1923), 1274-1277.
  • [2] A. V. Arhangel'skiĭ, On the metrization of topological spaces, Bull. Acad. Polon. Math. Ser. 8 (1960), 589-595. (Russian) MR 0125561 (23:A2861)
  • [3] -, Some metrization theorems, Uspehi Mat. Nauk 18 (1963), no. 5 (113), 139-145. (Russian) MR 0156318 (27:6242)
  • [4] D. K. Burke, Pseudo-open mappings from topological sums, Proc. Amer. Math. Soc. 74 (1979), 191-196. MR 521897 (80h:54034)
  • [5] J. Chaber, M. M. Čoban, and K. Nagami, On monotonic generalizations of Moore spaces, Čech complete spaces and $ p$-spaces, Fund. Math. 84 (1974), 107-119. MR 0343244 (49:7988)
  • [6] P. J. Collins, G. M. Reed, A. W. Roscoe, and M. E. Rudin, A lattice of conditions on topological spaces, Proc. Amer. Math. Soc. 94 (1985), 487-496. MR 787900 (87b:54018)
  • [7] P. J. Collins and A. W. Roscoe, Criteria for metrisability, Proc. Amer. Math. Soc. 90 (1984), 631-640. MR 733418 (85c:54041)
  • [8] A. H. Frink, Distance functions and the metrization problem, Bull. Amer. Math. Soc. 43 (1937), 133-142. MR 1563501
  • [9] G. Gruenhage and P. Zenor, Proto-metrizable spaces, Houston J. Math. 3 (1977), 47-53. MR 0442895 (56:1270)
  • [10] R. 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 (45:2657)
  • [11] H. H. Hung, A contribution to the theory of metrization, Canad. J. Math. 29 (1977), 1145-1151. MR 0454929 (56:13172)
  • [12] -, On a theorem of Arhangel'skiĭ, Proc. Amer. Math. Soc. 82 (1981), 629-633.
  • [13] -, A refinement on Michael's characterization of paracompactness, Proc. Amer. Math. Soc. 83 (1981), 179-182. MR 620008 (82j:54031)
  • [14] J. L. Kelley, General topology, Van Nostrand, Princeton, N. J., 1955. MR 0070144 (16:1136c)
  • [15] H. W. Martin, A note on the Frink metrization theorem, Rocky Mountain J. Math. 6 (1976), 155-157. MR 0391035 (52:11857)
  • [16] E. Michael, The product of a normal space and a metric space need not be normal, Bull. Amer. Math. Soc. 69 (1963), 375-376. MR 0152985 (27:2956)
  • [17] J. Nagata, A contribution to the theory of metrization, J. Inst. Polytech. Osaka City Univ. 8 (1957), 185-192. MR 0097789 (20:4256)
  • [18] -, Modern general topology, 2nd rev. ed., North-Holland, Amsterdam, 1985. MR 831659 (87g:54003)
  • [19] R. H. Sorgenfrey, On the topological product of paracompact spaces, Bull. Amer. Math. Soc. 53 (1947), 631-632. MR 0020770 (8:594f)
  • [20] J. M. Worrell Jr. and H. H. Wicke, Characterizations of developable topological spaces, Canad. J. Math. 17 (1965), 820-830. MR 0182945 (32:427)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1987-0908667-X
Keywords: Metrizability without uniformity among members of bases, pairnetworks, nests, companions of nests, capture of nests by their companions
Article copyright: © Copyright 1987 American Mathematical Society

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