Skip to main content

Cauchy sequences in quasi-pseudo-metric spaces


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.


  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



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).

Download citation


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