SpringerLink
Log in

Some results in metrization theory, 1950–1972

  • Conference paper
  • First Online:
Topology Conference

Part of the book series: Lecture Notes in Mathematics ((LNM,volume 375))

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution

Subscribe and save

Springer+ Basic
EUR 32.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or Ebook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 34.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 46.00
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free ship** worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Bibliography

  1. C. C. Alexander, An extension of Morita's metrization theorem, Proc. Amer. Math. Soc., 30 (1971), 578–581.

    MathSciNet  MATH  Google Scholar 

  2. P. S. Alexandroff and P. Urysohn, Une condition nessaire et suffisante pour qu'une class (L) soit une class (D), C. R. Acad. Aci. Paris, 177 (1923), 1274–1277.

    MATH  Google Scholar 

  3. P. S. Alexandroff, On some results in the theory of topological spaces, obtained within the last twenty-five years, Russian Math. Surveys, 15 (1960), 28–83.

    MathSciNet  Google Scholar 

  4. ______________, On some basic directions in general topology, Russian Math. Surveys, 19 (1964), 1–39.

    Google Scholar 

  5. ______________, On the metrization of topological spaces, Bull. Acad. Polon. Math. Ser. 8 (1960), 135–140 (in Russian).

    MathSciNet  Google Scholar 

  6. A. V. Arhangel'skii, New criteria for paracompactness and metrizability of an arbitrary T1-space, Soviet Math. Dokl., 2 (1961), 1367–1369.

    Google Scholar 

  7. ______________, On a class of spaces containing all metric and all locally bicompact spaces, Soviet Math. Dokl., 4 (1963), 1051–1055.

    MathSciNet  MATH  Google Scholar 

  8. ______________, On the metrization of topological spaces, Bull. Acad. Polon. Math. Ser. 8 (1960), 589–595 (in Russian).

    MathSciNet  Google Scholar 

  9. ______________, Some metrization theorems, Uspehi Mat. Nauk 18 (1963), no. 5 (113), 139–145 (in Russian).

    MathSciNet  Google Scholar 

  10. ______________, Map**s and spaces, Russian Math. Surveys, 21 (1966), 115–162.

    Article  MathSciNet  Google Scholar 

  11. C. E. Aull, A note on countably paracompact spaces and metrization, Proc. Amer. Math. Soc., 16 (1965), 1316–1317.

    Article  MathSciNet  MATH  Google Scholar 

  12. _____, Quasi-developments and δθ-bases, J. London Math Soc., to appear.

    Google Scholar 

  13. _____, Some base axioms for topology involving enumerability, General Topology and Its Relations to Modern Analysis and Algebra III, Academic Press (1971), 55–61.

    Google Scholar 

  14. __________, Topological spaces with a σ-point finite base, Proc. Amer. Math. Soc., 29 (1971) 411–416.

    MathSciNet  MATH  Google Scholar 

  15. H. R. Bennett, A note on the metrizability of M-spaces, Proc. Japan Acad. 45 (1969), 6–9.

    Article  MathSciNet  MATH  Google Scholar 

  16. __________, On quasi-developable spaces, Gen. Top. Appl., 1 (1971), 253–262.

    Article  MathSciNet  MATH  Google Scholar 

  17. __________ and D. J. Lutzer, A note on weak θ-refinability, Gen. Top. Appl., 2 (1972), 49–54.

    Article  MathSciNet  MATH  Google Scholar 

  18. __________, A note on point-countability in linearly ordered spaces, Proc. Amer. Math. Soc., 28 (1971), 598–606.

    Article  MathSciNet  MATH  Google Scholar 

  19. _____ and E. S. Berney, On Certain generalizations of developable spaces, (To Appear).

    Google Scholar 

  20. R. H. Bing, Metrization of topological spaces, Canad. J. Math., 3 (1951), 175–186.

    Article  MathSciNet  MATH  Google Scholar 

  21. C. J. R. Borges, On metrizability of topological spaces, Canad. J. Math., 20 (1968), 795–804.

    Article  MathSciNet  MATH  Google Scholar 

  22. __________, On stratifiable spaces, Pac. J. Math., 17 (1966), 1–16.

    Article  MathSciNet  MATH  Google Scholar 

  23. J. R. Boyd, Axioms that define semi-metric, Moore, and metric spaces, Proc. Amer. Math. Soc., 13 (1962), 482–484.

    MathSciNet  MATH  Google Scholar 

  24. D. K. Burke, On p-spaces and wΔ-spaces, Pac. J. Math., 35 (1970), 285–296.

    Article  MathSciNet  MATH  Google Scholar 

  25. _____, Spaces with a Gδ-diagonal, Proceedings TOPO ′72, (To Appear).

    Google Scholar 

  26. __________ and R. A. Stoltenberg, A note on p-spaces and Moore spaces, Pac. J Math., 30 (1969), 601–608.

    Article  MathSciNet  MATH  Google Scholar 

  27. J. G. Ceder, Some generalizations of metric spaces, Pac. J. Math., 11 (1961), 105–126.

    Article  MathSciNet  MATH  Google Scholar 

  28. E. W. Chittenden, On the equivalence of écart and voisinage, Trans. Amer. Math. Soc., 18 (1917), 161–166.

    MathSciNet  MATH  Google Scholar 

  29. __________, On the metrization problem and related problems in the theory of abstract sets, Bull. Amer. Math. Soc., 33 (1927), 13–34.

    Article  MathSciNet  MATH  Google Scholar 

  30. M. M. Čoban, Some metrization theorems for p-spaces, Soviet Math. Dokl., 8 (1967), 561–563.

    Google Scholar 

  31. H. H. Corson and E. Michael, Metrizability of certain countable unions, Ill. J. Math., 8 (1964), 351–360.

    MathSciNet  MATH  Google Scholar 

  32. G. D. Creede, Concerning semi-stratifiable spaces, Pac. J. Math., 32 (1970), 47–54.

    Article  MathSciNet  MATH  Google Scholar 

  33. V. V. Filippov, On feathered paracompacta, Soviet Math. Dokl., 9 (1968), 161–164.

    MATH  Google Scholar 

  34. S. P. Franklin, A new metrization proof, General Topology and Its Relations to Modern Analysis and Algebra III, Academic Press (1971), 127–130.

    Google Scholar 

  35. M. Fréchet, Sur Quelques Points du Calcul Fonctionnel, Rendiconti di Palermo, 22 (1906), 1–74.

    Article  MATH  Google Scholar 

  36. A. H. Frink, Distance functions and the metrization problem, Bull. Amer. Math. Soc., 43 (1937), 133–142.

    Article  MathSciNet  MATH  Google Scholar 

  37. E. E. Grace and R. W. Heath, Separability and metrizability in pointwise paracompact Moore spaces, Duke Math. J., 31 (1964), 603–610.

    Article  MathSciNet  MATH  Google Scholar 

  38. R. W. Heath, On spaces with point-countable bases, Bull. Acad. Polon. Sci. Sér. Math. Astronom. Phys., 13 (1965), 393–395.

    MathSciNet  MATH  Google Scholar 

  39. __________, On open map**s and certain spaces satisfying the first countability axiom, Fund. Math., 57 (1965), 91–96.

    MathSciNet  MATH  Google Scholar 

  40. __________, Arc-wise connectedness in semi-metric spaces, Pac. J. Math., 12 (1962), 1301–1319.

    Article  MathSciNet  MATH  Google Scholar 

  41. _____, Semi-metric spaces and related spaces, Topology Conference, Arizona State University (1967), 153–161.

    Google Scholar 

  42. _____, On p-spaces, q-spaces, r-spaces, and "s"-spaces, Proceedings of the Auburn Topology Conference (1969), 123–134.

    Google Scholar 

  43. __________, On certain first countable spaces, Topology Seminar, Wisconsin, 1965, Annals of Mathematical Studies, 60, edited by R. H. Bing and R. J. Bean, Princeton (1966), 103–113.

    Google Scholar 

  44. __________, Screenability, pointwise paracompactness, and metrization of Moore spaces, Canad. J. Math., 16 (1964), 763–770.

    Article  MathSciNet  MATH  Google Scholar 

  45. _____ and R. E. Hodel, Characterization of σ-spaces, Fund. Math, (To Appear)

    Google Scholar 

  46. R. E. Hodel, Moore spaces and wΔ-spaces, Pac. J. Math., 38 (1971), 641–652.

    Article  MathSciNet  MATH  Google Scholar 

  47. _____, Spaces characterized by sequences of covers which guarantee that certain sequences have cluster points, Proceedings of the University of Houston Point Set Topology Conference (1971), 105–114.

    Google Scholar 

  48. __________, Spaces defined by sequences of open covers which guarantee that certain sequences have cluster points, Duke Math. J., 39 (1972), 253–263.

    Article  MathSciNet  MATH  Google Scholar 

  49. T. Ishii, On wM-spaces I, II, Proc. Japan Acad., 46 (1970), 5–15.

    Article  MathSciNet  MATH  Google Scholar 

  50. __________ and T. Shiraki, Some properties of wM-spaces, Proc. Japan Acad., 47 (1971), 167–172.

    Article  MathSciNet  MATH  Google Scholar 

  51. F. B. Jones, Metrization, Amer. Math. Monthly, 73 (1966), 571–576.

    Article  MathSciNet  MATH  Google Scholar 

  52. __________, R. L. Moore's Axiom 1' and Metrization, Proc. Amer. Math. Soc., 9 (1958), 487.

    Article  MathSciNet  MATH  Google Scholar 

  53. D. E. Kullman, Developable spaces and p-spaces, Proc. Amer. Math. Soc., 27 (1971), 154–160.

    MathSciNet  MATH  Google Scholar 

  54. D. J. Lutzer, A metrization theorem for linearly orderable spaces, Proc. Amer. Math. Soc., 22 (1969), 557–558.

    MathSciNet  MATH  Google Scholar 

  55. _____, On quasi-uniform bases, Proceedings of the University of Oklahoma Topology Conference (1972), 104–117.

    Google Scholar 

  56. L. F. McAuley, A relation between perfect separability, completeness, and normality in semi-metric spaces, Pac. J. Math., 6 (1956), 315–326.

    Article  MathSciNet  MATH  Google Scholar 

  57. E. Michael, Spaces with point-countable bases, Proceedings of the University of Houston Point Set Topology Conference (1971), 37–41.

    Google Scholar 

  58. __________, A note on paracompact spaces, Proc. Amer. Math. Soc., 4 (1953), 831–838.

    Article  MathSciNet  MATH  Google Scholar 

  59. _____, On Nagami's Σ-spaces and some related matters, Proceedings of the Washington State University Conference on General Topology (1970), 13–19.

    Google Scholar 

  60. __________, ℵ0-spaces, J. Math. Mech., 15 (1966), 983–1002.

    MathSciNet  Google Scholar 

  61. __________, Yet another note on paracompact spaces, Proc. Amer. Math. Soc., 10 (1959), 309–314.

    Article  MathSciNet  MATH  Google Scholar 

  62. _____ and F. G. Slaughter, Jr., Σ-spaces with point-countable separating open covers are σ-spaces, unpublished.

    Google Scholar 

  63. A. S. Miščenko, Spaces with point-countable bases, Soviet Math. Dokl., 3 (1962), 855–858.

    Google Scholar 

  64. R. L. Moore, A set of axioms for plane analysis situs, Fund. Math. 25 (1935), 13–28.

    MATH  Google Scholar 

  65. K. Morita, Products of normal spaces with metric spaces, Math. Ann., 154 (1964), 365–382.

    Article  MathSciNet  MATH  Google Scholar 

  66. __________, On the simple extension of a space with respect to a uniformity, IV, Proc. Japan Acad., 27 (1951), 632–636.

    Article  MathSciNet  MATH  Google Scholar 

  67. __________, A condition for the metrizability of topological spaces and for n-dimensionality, Sci. Rep. Tokyo Kyoiki Daigaku, Ser. A, 5 (1955), 33–36.

    MathSciNet  MATH  Google Scholar 

  68. K. Nagami, Σ-spaces, Fund. Math., 65 (1969), 169–192.

    MathSciNet  MATH  Google Scholar 

  69. __________, σ-spaces and product spaces, Math. Ann., 181 (1969), 109–118.

    Article  MathSciNet  MATH  Google Scholar 

  70. __________, Note on metrizability and n-dimensionality, Proc. Japan Acad., 36 (1960), 565–570.

    Article  MathSciNet  MATH  Google Scholar 

  71. J. Nagata, Theorem for metrizability of topological spaces, Proc. Japan Acad., 33 (1957), 128–130.

    Article  MathSciNet  MATH  Google Scholar 

  72. __________, Note on a theorem for metrization, Proc. Japan Acad., 33 (1957), 613–615.

    Article  MathSciNet  Google Scholar 

  73. __________, On a necessary and sufficient condition of metrizability, J. Inst. Polytech. Osaka City Univ. Ser. A, Math., 1 (1950), 93–100.

    MathSciNet  Google Scholar 

  74. __________, A contribution to the theory of metrization, J. Inst. Polytech. Osaka City Univ. Ser. A, 8 (1957), 185–192.

    MathSciNet  Google Scholar 

  75. _____, Modern General Topology, Amsterdam-Groningen (1968).

    Google Scholar 

  76. __________, A note on Filipov's theorem, Proc. Japan Acad., 45 (1969), 30–33.

    Article  MathSciNet  MATH  Google Scholar 

  77. _____, Problems on generalized metric spaces II, Proceedings of the University of Houston Point Set Topology Conference (1971), 42–52.

    Google Scholar 

  78. __________ and F. Siwiec, A note on nets and metrization, Proc. Japan Acad., 44 (1968), 623–627.

    Article  MathSciNet  MATH  Google Scholar 

  79. V. W. Niemytski, On the "third axiom of metric spaces," Trans. Amer. Math. Soc., 29 (1927), 507–513.

    MathSciNet  Google Scholar 

  80. L. J. Norman, A sufficient condition for the quasi-metrizability of a topological space, Port. Math., 26 (1967), 207–211.

    MathSciNet  MATH  Google Scholar 

  81. A. Okuyama, On metrizability of M-spaces, Proc. Japan Acad., 40 (1964), 176–179.

    Article  MathSciNet  MATH  Google Scholar 

  82. __________, Some generalizations of metric spaces, their metrization theorems and product theorems, Sci. Rep. Tokyo Kyoiku Daigaku, Ser. A, 9 (1967), 236–254.

    Google Scholar 

  83. P. A. O'Meara, A metrization theorem, Math. Nach., 45 (1970), 69–72.

    Article  MathSciNet  MATH  Google Scholar 

  84. V. I. Ponomarev, Axioms of countability and continuous map**s, Bull. Acad. Polon. Sci. Ser. Math., 8 (1960), 127–134 (in Russian).

    MathSciNet  MATH  Google Scholar 

  85. __________, Metrizability of a finally compact p-space with a point-countable base, Soviet Math. Dokl., 8 (1967), 765–768.

    MATH  Google Scholar 

  86. G. M. Reed, On screenability and metrizability of Moore spaces, Canad. J. Math., 13 (1971), 1087–1092.

    Article  MathSciNet  MATH  Google Scholar 

  87. H. Ribeiro, Sur les espaces a' metrique faible, Port. Math., 4 (1943), 21–40 and 65–68.

    MathSciNet  MATH  Google Scholar 

  88. D. Rolfsen, Alternate metrization proofs, Canad. J. Math., 18 (1966), 750–757.

    Article  MathSciNet  MATH  Google Scholar 

  89. T. Shiraki, M-spaces, their generalization and metrization theorems, Sci. Rep. Tokyo Kyoiku Daigaku, Ser. A, 11 (1971), 57–67.

    MathSciNet  MATH  Google Scholar 

  90. M. Sion and G. Zelmer, On quasi-metrizability, Canad. J. Math., 19 (1967), 1243–1249.

    Article  MathSciNet  MATH  Google Scholar 

  91. F. Siwiec, Metrizability of M-spaces, Notices Amer. Math. Soc., 15 (1968), 224-#68T-111.

    Google Scholar 

  92. Yu M. Smirnov, A necessary and sufficient condition for metrizability of a topopolical space, Doklady Akad. Nauk SSSR (N.S.), 77 (1951), 197–200 (in Russian).

    MathSciNet  MATH  Google Scholar 

  93. J. C. Smith and L. L. Krajewski, Expandability and collectionwise normality, Trans. Amer. Math. Soc., 160 (1971), 437–451.

    MathSciNet  MATH  Google Scholar 

  94. V. E. Šneider, Continuous images of Suslin and Borel sets. Metrization theorems. Dokl. Acad. Nauk SSSR, 50 (1945), 77–79.

    MathSciNet  Google Scholar 

  95. R. H. Sorgenfrey, On the topological product of paracompact spaces, Bull. Amer. Math. Soc., 53 (1947), 631–632.

    Article  MathSciNet  MATH  Google Scholar 

  96. L. A. Steen, Conjectures and counterexamples in metrization theory, Amer. Math. Monthly, 79 (1972), 113–132.

    Article  MathSciNet  MATH  Google Scholar 

  97. R. A. Stoltenberg, On quasi-metric spaces, Duke Math. J., 36 (1969), 65–72.

    Article  MathSciNet  MATH  Google Scholar 

  98. A. H. Stone, Sequences of coverings, Pac. J. Math., 10 (1960), 689–691.

    Article  MathSciNet  MATH  Google Scholar 

  99. __________, Paracompactness and product spaces, Bull. Amer. Math. Soc., 54 (1948), 977–982.

    Article  MathSciNet  MATH  Google Scholar 

  100. S. Swaminathan, Elasticity, framing and paracompactness, Proceedings TOPO '72, (To Appear).

    Google Scholar 

  101. F. Tall, Some set-theoretic consistency results in topology, (To Appear).

    Google Scholar 

  102. D. R. Traylor, Concerning metrizability of pointwise paracompact Moore spaces, Canad. J. Math., 16 (1964), 407–411.

    Article  MathSciNet  MATH  Google Scholar 

  103. __________, A note on metrizability of Moore spaces, Proc. Amer. Math. Soc., 14 (1963), 804–805.

    Article  MathSciNet  MATH  Google Scholar 

  104. A. Tychonoff, Über einen Metrisationssatz von P. Urysohn, Math. Ann., 95 (1926), 139–142.

    Article  MathSciNet  MATH  Google Scholar 

  105. P. Urysohn, Zum Metrisationsproblem, Math. Ann., 94 (1925), 309–315.

    Article  MathSciNet  MATH  Google Scholar 

  106. C. W. Vickery, Axioms for Moore spaces and metric spaces, Bull. Amer. Math. Soc., 46 (1940), 560–564.

    Article  MathSciNet  MATH  Google Scholar 

  107. H. H. Wicke and J. M. Worrell Jr., Characterizations of developable spaces, Canad. J. Math., 17 (1965), 820–830.

    Article  MathSciNet  MATH  Google Scholar 

  108. W. A. Wilson, On semi-metric spaces, Amer. J. Math., 53 (1931), 361–373.

    Article  MathSciNet  MATH  Google Scholar 

  109. __________, On quasi-metrics, Amer. J. Math., 53 (1931), 675–684.

    Article  MathSciNet  MATH  Google Scholar 

  110. M. R. Wiscamb, On symmetric neighborhood systems in metric, strongly paracompact and some other types of spaces, Trans. Amer. Math. Soc., 137 (1969), 431–450.

    Article  MathSciNet  MATH  Google Scholar 

  111. P. Zenor, On spaces with regular Gδ-diagonals, Pac. J. Math., 40 (1972), 759–763.

    Article  MathSciNet  MATH  Google Scholar 

Download references

Authors
  1. R. E. Hodel
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Raymond F. Dickman Jr. Peter Fletcher

Rights and permissions

Reprints and permissions

Copyright information

© 1974 Springer-Verleg

About this paper

Cite this paper

Hodel, R.E. (1974). Some results in metrization theory, 1950–1972. In: Dickman, R.F., Fletcher, P. (eds) Topology Conference. Lecture Notes in Mathematics, vol 375. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0064018

Download citation

  • DOI: https://doi.org/10.1007/BFb0064018

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-06684-2

  • Online ISBN: 978-3-540-37948-5

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation