Skip to main content
SpringerLink
Log in

Modifications of uniform bases and classification of topological spaces

  • Published:
Journal of Mathematical Sciences Aims and scope Submit manuscript

Abstract

The notion of a uniform base, which was introduced by P. S. Alexandroff in 1960, turned out to be deeply connected with various properties of topological spaces. Modifications of this notion have led to new metrizability criteria and new directions in classification of topological spaces. A survey of the related results is given.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. P. S. Aleksandrov, “On metrization of topological spaces,” Bull. Polish Acad. Sci., Ser. Mat., 8, No. 3, 135–140 (1960).

    MATH  Google Scholar 

  2. B. Alleche, A. Arhangel’skii, and J. Calbrix, “Weak developments and metrization,” Topology Appl., 100, 23–38 (2000).

    Article  MATH  MathSciNet  Google Scholar 

  3. A. V. Arkhangel’skii, “Some metrization theorems,” Usp. Mat. Nauk, 18, No. 5, 139–145 (1963).

    MathSciNet  Google Scholar 

  4. A. V. Arkhangel’skii, “Mappings and spaces,” Usp. Mat. Nauk, 21, No. 4, 133–184 (1966).

    MathSciNet  Google Scholar 

  5. A. V. Arkhangel’skii, “Compactness,” in: Itogi Nauki Tekh. Sovr. Probl. Mat. Fundam. Napr., 50, VINITI, Moscow (1989), pp. 5–128.

    Google Scholar 

  6. A. V. Arkhangel’skii, “Paracompactness and metrization. The method of covers in classification of spaces,” in: Itogi Nauki Tekhn. Sovr. Probl. Mat. Fundam. Napr., 51, VINITI, Moscow (1989), pp. 5–80.

    Google Scholar 

  7. A. V. Arhangel’skii, W. Just, E. Reznichenko, and P. J. Szeptyczki, “Sharp bases and weakly uniform bases versus point countable bases,” Topology Appl., 100, 39–46 (2000).

    Article  MATH  MathSciNet  Google Scholar 

  8. B. Bailey and G. Gruenhage, “On a question concerning sharp bases” (to appear).

  9. Z. Balogh, S. Davis, W. Just, S. Shelah, and P. J. Szeptyczki, “Strongly almost disjoint sets and weakly uniform bases,” Trans. Amer. Math. Soc., 352, 4971–4987 (2000).

    Article  MATH  MathSciNet  Google Scholar 

  10. Z. Balogh and G. Gruenhage, “Base multiplicity in compact and generalized compact spaces,” Topology Appl., 115, No. 2, 139–151 (2001).

    Article  MATH  MathSciNet  Google Scholar 

  11. H. Bennet and R. Berney, “Subparacompactness and G δ-diagonals,” Proc. Amer. Math. Soc., 30, No. 3 (1971).

  12. H. Bennett and D. Lutzer, “Ordered spaces with special bases,” Fund. Math., 158, 289–299 (1998).

    MATH  MathSciNet  Google Scholar 

  13. H. Bennett and D. Lutzer, “Spaces with <ω-weakly uniform bases,” Topology Proc., 27, No. 1, 39–49 (2003).

    MATH  MathSciNet  Google Scholar 

  14. R. H. Bing, “Metrization of topological spaces,” Can. J. Math., 3, 175–186 (1951).

    MATH  MathSciNet  Google Scholar 

  15. P. A. Biryukov, “Ranks of families of sets and properties of topological spaces,” Dokl. Akad. Nauk SSSR, 257, 777–779 (1981).

    MathSciNet  Google Scholar 

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

    MathSciNet  Google Scholar 

  17. S. W. Davis, G. M. Reed, and M. L. Wage, “Further results on weakly uniform bases,” Houston J. Math., 2, No. 1, 57–63 (1976).

    MATH  MathSciNet  Google Scholar 

  18. R. Engelking, General Topology, Heldermann, Berlin (1989).

    MATH  Google Scholar 

  19. P. Erdős and R. Rado, “A partition calculus in set theory,” Bull. Amer. Math. Soc., 62, 427–489 (1956).

    Article  MathSciNet  Google Scholar 

  20. G. Gruenhage, “Generalized metric spaces,” in: Handbook of Set-Theoretic Topology (K. Kunen and J. E. Vaughan, eds.), North-Holland, Amsterdam (1984), pp. 425–501.

    Google Scholar 

  21. G. Gruenhage and P. Nyikos, “Spaces with bases of countable rank,” Gen. Top. Appl., 8, 233–257 (1978).

    Article  MathSciNet  Google Scholar 

  22. A. Hajnal and I. Juhasz, “On hereditarily α-Lindelöf and hereditarily α-separable spaces,” Ann. Univ. Sci. Budapest. Eőtvős Sect. Math., 11, 115–124 (1968).

    MATH  MathSciNet  Google Scholar 

  23. Handbook of Mathematical Logic. Part II. Set Theory (K. J. Barwise, ed.), North-Holland, Amsterdam (1977).

    Google Scholar 

  24. R. W. Heath and W. E. Lindgren, “Weakly uniform bases,” Houston J. Math., 2, No. 1, 85–90 (1976).

    MATH  MathSciNet  Google Scholar 

  25. I. Juhasz, “A survey of S and L spaces,” Colloq. Math. Soc. Janos Bolyai, 23, 675–688 (1978).

    MathSciNet  Google Scholar 

  26. I. Juhasz, A survey of S and L spaces, Preprint No. 15/1979, Math. Inst. Hungarian Acad. Sci (1979).

  27. K. Kuratowski, Topology, Vol. 1, Academic Press, New York-London; PWN, Warsaw (1966).

    Google Scholar 

  28. E. Michael, “Point-finite and locally finite coverings,” Can. J. Math., 7, 275–279 (1955).

    MATH  MathSciNet  Google Scholar 

  29. E. Michael, “Paracompactness and the Lindelöf property in finite and countable Cartesian products,” Comp. Math., 23, 199–214 (1971).

    MATH  MathSciNet  Google Scholar 

  30. A. S. Mishchenko, “Spaces with point-countable bases,” Dokl. Akad. Nauk SSSR, 144, 985–988 (1962).

    MathSciNet  Google Scholar 

  31. S. A. Peregudov, “On some properties of families of open sets and covers,” Vestn. Mosk. Univ., Ser. 1: Mat. Mekh., 3, 25–33 (1976).

    MathSciNet  Google Scholar 

  32. S. A. Peregudov, “Metrizability in the class of topological spaces with weakly uniform bases,” Bull. Polish Akad. Sci., Ser. Math., 28, Nos. 11–12, 609–612 (1980).

    MATH  MathSciNet  Google Scholar 

  33. S. A. Peregudov, “On some generalizations of the notion of a uniform base,” in: Mappings and Functors [in Russian], Moscow (1984), pp. 102–117.

  34. S. A. Peregudov, “Weakly uniform bases and the first axiom of countability,” Mat. Zametki, 40, No. 3, 331–340 (1986).

    MathSciNet  Google Scholar 

  35. S. A. Peregudov, “On the Noetherian type of topological spaces,” Comment. Math. Univ. Carolin., 38, No. 3, 581–586 (1997).

    MATH  MathSciNet  Google Scholar 

  36. S. A. Peregudov, “On pseudocompactness and other covering properties,” Questions Answers Gen. Topology, 17, 153–155 (1999).

    MATH  MathSciNet  Google Scholar 

  37. S. A. Peregudov, “On Boolean algebras of many-valued mappings,” Topology Proc., 24, 407–419 (1999).

    MATH  MathSciNet  Google Scholar 

  38. S. A. Peregudov, “On n-in-countable bases,” Comment. Math. Univ. Carolin., 41, No. 1, 175–178 (2000).

    MATH  MathSciNet  Google Scholar 

  39. S. A. Peregudov and B. E. Shapirovskii, “A class of compact spaces,” Dokl. Akad. Nauk SSSR, 230, 279–282 (1976).

    MathSciNet  Google Scholar 

  40. C. Pixley and P. Roy, “Uncompletable Moore spaces,” in: Proc. Auburn. Topol. Conf. (1969), pp. 75–85.

  41. V. I. Ponomarev, “On metrizability of Lindelof spaces with point-countable bases,” Dokl. Akad. Nauk SSSR, 174, 1274–1277 (1967).

    MathSciNet  Google Scholar 

  42. V. V. Popov, “A perfect map need not preserve a G δ-diagonal,” Gen. Top. Appl., 7, 31–33 (1977).

    Article  Google Scholar 

  43. D. B. Shakhmatov, “On pseudocompact spaces with point-countable bases,” Dokl. Akad. Nauk SSSR, 279, No. 5, 825–829 (1984).

    MathSciNet  Google Scholar 

  44. D. B. Shakhmatov, “Compact spaces and their generalizations,” in: Recent Progress in General Topology (M. Husek and J. van Mill, eds.), North-Holland, Amsterdam (1992), pp. 573–640.

    Google Scholar 

  45. Yu. M. Smirnov, “On the theory of Lindelöf spaces,” Ukr. Mat. Zh., 3, No. 1, 52–60 (1951).

    MATH  Google Scholar 

  46. Z. Szentmiklossy, “S-Spaces and L-spaces under Martin’s axiom,” Topology Appl., 16, No. 3, 243–251 (1983).

    Article  MATH  MathSciNet  Google Scholar 

  47. V. V. Uspenskii, “Pseudocompact spaces with a σ-point-finite base,” Comment. Math. Univ. Carolin., 25, 261–264 (1984).

    MathSciNet  Google Scholar 

  48. N. V. Velichko, “On the cardinality of open covers of topological spaces,” Fund. Math., 53, 271–282 (1973).

    Google Scholar 

  49. H. Wicke and J. Worrell, “Characterizations of developable topological spaces,” Can. J. Math., 17, 820–830 (1965).

    MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Applied Mathematics, State University of Management, Russia

    S. A. Peregudov

Authors
  1. S. A. Peregudov
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

__________

Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 34, General Topology, 2005.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Peregudov, S.A. Modifications of uniform bases and classification of topological spaces. J Math Sci 144, 4184–4204 (2007). https://doi.org/10.1007/s10958-007-0260-9

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10958-007-0260-9

Keywords

Search

Navigation