Skip to main content

Cauchy sequences in quasi-pseudo-metric spaces

Abstract

This paper considers the problem of defining Cauchy sequence and completeness in quasi-pseudo-metric spaces. The definitions proposed allow versions of such classical theorems as the Baire Category Theorem, the Contraction Principle and Cantor's characterization of completeness to be formulated in the quasi-pseudo-metric setting.

This is a preview of subscription content, access via your institution.

References

  1. [1]

    Albert, G. E.: A note on quasi-metric spaces. Bull. Amer. Math. Soc.47, 479–482 (1941).

    Google Scholar 

  2. [2]

    Balanzat, M.: Sobre la metrización de los espacios cuasi métricos. Gaz. Mat. Lisboa50, 91–94 (1951).

    Google Scholar 

  3. [3]

    di Concilio, A.: Spazi quasimetrici e topologie ad essi associate. Rend. Fis. Mat. Napoli38, 113–130 (1971).

    Google Scholar 

  4. [4]

    Domiaty, R. Z.: The Hausdorff separation property for space-time. Eleftheria (Athens). (In print.)

  5. [5]

    Domiaty, R. Z.: Life withoutT 2. Differential-Geometric Methods in Theoretical Physics. Conf. Clausthal-Zellerfeld (FRG), July 1978. Lecture Notes Physics. Berlin-Heidelberg-New York: Springer. 1980.

    Google Scholar 

  6. [6]

    Dutta, M., Das, M. K., Majumdar, M.: On some generalization of fixed point theorems with applications in operator equations. Glasnik Mat.9(29), 155–159 (1974).

    Google Scholar 

  7. [7]

    Fletcher, P., Lindgren, W. F.: Transitive quasi-uniformities. J. Math. Anal. Appl.39, 397–405 (1972).

    Google Scholar 

  8. [8]

    Heath, R. W.: A note on quasi-metric spaces. Notices Amer. Math. Soc.18, 786 (1971).

    Google Scholar 

  9. [9]

    Kelly, J. C.: Bitopological spaces. Proc. London Math. Soc.13, 71–89 (1963).

    Google Scholar 

  10. [10]

    Kofner, J. A.: On Δ-metrizable spaces. Mat. Zametki13, 277–287 (1972).

    Google Scholar 

  11. [11]

    Nedev, S. I., Choban, M. M.: On the theory of 0-metrizable spaces, I, II, III. Vestnik Moskov. Univ. Ser. I Mat. Meh.27, # 1, 8–15; # 2, 10–17; # 3, 10–15 (1972).

    Google Scholar 

  12. [12]

    Patty, C. W.: Bitopological spaces. Duke Math. J.34, 387–391 (1967).

    Google Scholar 

  13. [13]

    Reilly, I. L.: Quasi-gauge spaces. J. London Math. Soc. (2)6, 481–487 (1973).

    Google Scholar 

  14. [14]

    Reilly, I. L.: A generalized contraction principle. Bull. Austral. Math. Soc.10, 359–363 (1974).

    Google Scholar 

  15. [15]

    Ribeiro, H.: Sur les espaces à métrique faible. Portugaliae Math.4, 21–40 (1943).

    Google Scholar 

  16. [16]

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

    Google Scholar 

  17. [17]

    Stoltenberg, R. A.: On quasi-metric spaces. Duke Math. J.36, 65–71 (1969).

    Google Scholar 

  18. [18]

    Subrahmanyam, P. V.: Remarks on some fixed point theorems related to Banach's contraction principle. J. Math. Phys. Sci.8, 455–457 (1974).

    Google Scholar 

  19. [19]

    Tan, K. K.: Fixed point theorems for nonexpansive mappings. Pac. J. Math.41, 829–842 (1972).

    Google Scholar 

  20. [20]

    Waterman, M. S., Smith, T. F., Beyer, W. A.: Some biological sequence metrics. Advances Math.20, 367–387 (1976).

    Google Scholar 

  21. [21]

    Wilson, W. A.: On quasi-metric spaces. Amer. J. Math.53, 675–684 (1931).

    Google Scholar 

Download references

Author information

Affiliations

Authors

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Reilly, I.L., Subrahmanyam, P.V. & Vamanamurthy, M.K. Cauchy sequences in quasi-pseudo-metric spaces. Monatshefte für Mathematik 93, 127–140 (1982). https://doi.org/10.1007/BF01301400

Download citation

Keywords

  • Cauchy Sequence
  • Classical Theorem
  • Contraction Principle
  • Category Theorem
  • Baire Category