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.
Similar content being viewed by others
References
Albert, G. E.: A note on quasi-metric spaces. Bull. Amer. Math. Soc.47, 479–482 (1941).
Balanzat, M.: Sobre la metrización de los espacios cuasi métricos. Gaz. Mat. Lisboa50, 91–94 (1951).
di Concilio, A.: Spazi quasimetrici e topologie ad essi associate. Rend. Fis. Mat. Napoli38, 113–130 (1971).
Domiaty, R. Z.: The Hausdorff separation property for space-time. Eleftheria (Athens). (In print.)
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.
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).
Fletcher, P., Lindgren, W. F.: Transitive quasi-uniformities. J. Math. Anal. Appl.39, 397–405 (1972).
Heath, R. W.: A note on quasi-metric spaces. Notices Amer. Math. Soc.18, 786 (1971).
Kelly, J. C.: Bitopological spaces. Proc. London Math. Soc.13, 71–89 (1963).
Kofner, J. A.: On Δ-metrizable spaces. Mat. Zametki13, 277–287 (1972).
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).
Patty, C. W.: Bitopological spaces. Duke Math. J.34, 387–391 (1967).
Reilly, I. L.: Quasi-gauge spaces. J. London Math. Soc. (2)6, 481–487 (1973).
Reilly, I. L.: A generalized contraction principle. Bull. Austral. Math. Soc.10, 359–363 (1974).
Ribeiro, H.: Sur les espaces à métrique faible. Portugaliae Math.4, 21–40 (1943).
Sion, M., Zelmer, G.: On quasi-metrizability. Canad. J. Math.19, 1243–1249 (1967).
Stoltenberg, R. A.: On quasi-metric spaces. Duke Math. J.36, 65–71 (1969).
Subrahmanyam, P. V.: Remarks on some fixed point theorems related to Banach's contraction principle. J. Math. Phys. Sci.8, 455–457 (1974).
Tan, K. K.: Fixed point theorems for nonexpansive mappings. Pac. J. Math.41, 829–842 (1972).
Waterman, M. S., Smith, T. F., Beyer, W. A.: Some biological sequence metrics. Advances Math.20, 367–387 (1976).
Wilson, W. A.: On quasi-metric spaces. Amer. J. Math.53, 675–684 (1931).
Author information
Authors and Affiliations
Rights 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
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF01301400