Surveys in Mathematics and its Applications

ISSN 1842-6298 (electronic), 1843-7265 (print)
 Volume 5 (2010), 321 -- 332

FUNCTION VALUED METRIC SPACES

Madjid Mirzavaziri

Abstract. In this paper we introduce the notion of an F-metric, as a function valued distance mapping, on a set X and we investigate the theory of F-metric spaces. We show that every metric space may be viewed as an F-metric space and every F-metric space (X,δ) can be regarded as a topological space (X,τδ). In addition, we prove that the category of the so-called extended F-metric spaces properly contains the category of metric spaces. We also introduce the concept of an `F-metric space as a completion of an F-metric space and, as an application to topology, we prove that each normal topological space is `F-metrizable.

2010 Mathematics Subject Classification: Primary 54E35; Secondary 54E70, 54A40, 46C05.
Keywords: Function valued metric; Positive element; Strictly positive element; F-completeness; F-metric space; Allowance set; F-Cauchy; F-completion; F-metrizable.

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Acknowledgement. This research was supported by a grant from Ferdowsi University of Mashhad; No. MP89156MIZ.

References

  1. T. Bag and S. K. Samanta, Finite dimensional fuzzy normed linear spaces, J. Fuzzy Math. 11(3) (2003), 687--705. MR2005663(2005k:46195). Zbl 1045.46048.

  2. T. Bag and S. K. Samanta, Fuzzy bounded linear operators, Fuzzy Sets and Systems, 151 (2005), 513--547. MR2126172 (2005m:47149). Zbl 1077.46059.

  3. S. C. Chang and J. N. Mordeson, Fuzzy linear operators and fuzzy normed linear spaces, Bull. Cal. Math. Soc., 86 (2004), 429--436. MR1351812(96e:46107). Zbl 0829.47063.

  4. C. Feblin, The completion of a fuzzy normed linear space, J. Math. Anal. Appl., 174 (1993), 428--440. MR1215623(94c:46159). Zbl 0806.46083.

  5. O. Kaleva and S. Seikkala, On fuzzy metric spaces, Fuzzy set and System, 12 (1984), 215--229. MR0740095(85h:54007). Zbl 0558.54003.

  6. A. K. Katsaras, Fuzzy topological vector space II, Fuzzy set and System, 12 (1984), 143--154. MR0734946(85g:46014).

  7. J. L. Kelley, General Topology, Graduate Texts in Mathematics No. 27. Springer-Verlag, New York-Berlin, 1975. MR0370454(51 #6681). Zbl 0306.54002.

  8. I. Kramosil and J. Michalek, Fuzzy metric and statistical metric spaces, Kybernetica, 11 (1975), 336--344. MR0410633(53 #14381). Zbl 0319.54002.

  9. E. C. Lance, Hilbert C*-modules. A toolkit for operator algebraists, LMS Lecture Note Series 210, Cambridge Univ. Press, 1995. MR1325694(96k:46100). Zbl 0822.46080.

  10. W. L. Paschke, Inner product modules over B*-algebras, Trans Amer. Math. Soc., 182 (1973), 443--468. MR0355613(50 #8087). Zbl 0239.46062.

  11. G. K. Pedersen, Analysis Now, Springer-Verlag New York, Inc., 1989. MR0971256(90f:46001). Zbl 0668.46002.

  12. I. Raeburn and D. P. Williams, Morita equivalence and continuous-trace C*-algebras, Mathematical surveys and Monographs AMS 60, 1998. MR1634408(2000c:46108). Zbl 0922.46050.

  13. B. Schweizer and A. Sklar, Probabilistic Metric Spaces, Elsevier Science Publishing Company, 1983. MR0790314(86g:54045). Zbl 0546.60010.

  14. N. E. Wegge-Olsen, K-Theory of C*-Algebras, Oxford Science Publications, 1993. MR1222415(95c:46116). Zbl 0780.46038.

  15. L. A. Zadeh, Fuzzy sets, Information and Control 8 (1965), 338--353. MR0219427(36 #2509). Zbl 0139.24606.



Madjid Mirzavaziri
Department of Pure Mathematics, Ferdowsi University of Mashhad,
P.O. Box 1159--91775, Iran.
and
Centre of Excellence in Analysis on Algebraic Structures (CEAAS),
Ferdowsi University of Mashhad, Iran.
e-mail: mirzavaziri@gmail.com, mirzavaziri@math.um.ac.ir

http://www.utgjiu.ro/math/sma