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Year 2019, Volume: 2 Issue: 3, 107 - 115, 30.09.2019
https://doi.org/10.32323/ujma.561120

Abstract

References

  • [1] M. Frechet, Sur quelques points du calcul fonctionnel, Rend. Circ. Mat. Palermo, 22 (1906), 1 - 74.
  • [2] E. Korczak-Kubiak, A. Loranty and R. J. Pawlak, Baire Generalized topological spaces, Generalized metric spaces and infinite Games, Acta Math. Hungar., 140 (2013), 203 - 231.
  • [3] K. Menger, statistical metrics, Proc. Nat. Acad. of Sci., U.S.A. 28 (1942), 535 - 537.
  • [4] K. Menger, Probabilistic theories of relations, Ibid., 37 (1951), 178 - 180.
  • [5] K. Menger, Probabilistic geometry, Ibid., 37 (1951), 226 - 229.
  • [6] W. K. Min, On weak neighborhood systems and spaces, Acta Math. Hungar., 121 (3) (2008), 283 - 292.
  • [7] B. Schweizer, A. Sklar and B. Thorp, The metrization of statistical metric spaces, Pacific J. Math., 10 (2) (1960), 673 - 675.
  • [8] E. O. Thorp, Best possible triangle inequalities for statistical metric spaces, Proc. Amer. Math. Soc., 11 (1960), 734 - 740.
  • [9] E. O. Thorp, Generalized topologies for statistical metric spaces, Fund. Math., Li, 51 (1962), 9 - 21.
  • [10] A. Wald, On a statistical generalization of metric spaces, Proc. Natl. Acad. Sci. USA., 29 (1943), 196 - 197.

On Various $g$-Topology in Statistical Metric Spaces

Year 2019, Volume: 2 Issue: 3, 107 - 115, 30.09.2019
https://doi.org/10.32323/ujma.561120

Abstract

The purpose of this paper is to analyze the significance of new $g$-topologies defined in statistical metric spaces and we prove various properties for the neighbourhoods defined by Thorp in statistical metric spaces. Also, we give a partial answer to the questions, namely "What are the necessary and sufficient conditions that the $g$-topology of $type V$ to be of $type V_{D}?,$ the $g$-topology of $type V_{\alpha}$ to be the $g$-topology of $type V_{D} ?$ and the $g$-topology of $type V_{\alpha}$ to be a topology?" raised by Thorp in 1962. Finally, we discuss the relations between $\M_{\Omega}$-open sets in generalized metric spaces and various $g$-topology neighbourhoods defined in statistical metric spaces. Also, we prove weakly complete metric space is equivalent to a complete metric space if $\Omega$ satisfies the $\mathcal{V}$-property. 

References

  • [1] M. Frechet, Sur quelques points du calcul fonctionnel, Rend. Circ. Mat. Palermo, 22 (1906), 1 - 74.
  • [2] E. Korczak-Kubiak, A. Loranty and R. J. Pawlak, Baire Generalized topological spaces, Generalized metric spaces and infinite Games, Acta Math. Hungar., 140 (2013), 203 - 231.
  • [3] K. Menger, statistical metrics, Proc. Nat. Acad. of Sci., U.S.A. 28 (1942), 535 - 537.
  • [4] K. Menger, Probabilistic theories of relations, Ibid., 37 (1951), 178 - 180.
  • [5] K. Menger, Probabilistic geometry, Ibid., 37 (1951), 226 - 229.
  • [6] W. K. Min, On weak neighborhood systems and spaces, Acta Math. Hungar., 121 (3) (2008), 283 - 292.
  • [7] B. Schweizer, A. Sklar and B. Thorp, The metrization of statistical metric spaces, Pacific J. Math., 10 (2) (1960), 673 - 675.
  • [8] E. O. Thorp, Best possible triangle inequalities for statistical metric spaces, Proc. Amer. Math. Soc., 11 (1960), 734 - 740.
  • [9] E. O. Thorp, Generalized topologies for statistical metric spaces, Fund. Math., Li, 51 (1962), 9 - 21.
  • [10] A. Wald, On a statistical generalization of metric spaces, Proc. Natl. Acad. Sci. USA., 29 (1943), 196 - 197.
There are 10 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Articles
Authors

V. Renukadevi 0000-0002-4185-5545

S. Vadakasi This is me

Publication Date September 30, 2019
Submission Date May 6, 2019
Acceptance Date August 3, 2019
Published in Issue Year 2019 Volume: 2 Issue: 3

Cite

APA Renukadevi, V., & Vadakasi, S. (2019). On Various $g$-Topology in Statistical Metric Spaces. Universal Journal of Mathematics and Applications, 2(3), 107-115. https://doi.org/10.32323/ujma.561120
AMA Renukadevi V, Vadakasi S. On Various $g$-Topology in Statistical Metric Spaces. Univ. J. Math. Appl. September 2019;2(3):107-115. doi:10.32323/ujma.561120
Chicago Renukadevi, V., and S. Vadakasi. “On Various $g$-Topology in Statistical Metric Spaces”. Universal Journal of Mathematics and Applications 2, no. 3 (September 2019): 107-15. https://doi.org/10.32323/ujma.561120.
EndNote Renukadevi V, Vadakasi S (September 1, 2019) On Various $g$-Topology in Statistical Metric Spaces. Universal Journal of Mathematics and Applications 2 3 107–115.
IEEE V. Renukadevi and S. Vadakasi, “On Various $g$-Topology in Statistical Metric Spaces”, Univ. J. Math. Appl., vol. 2, no. 3, pp. 107–115, 2019, doi: 10.32323/ujma.561120.
ISNAD Renukadevi, V. - Vadakasi, S. “On Various $g$-Topology in Statistical Metric Spaces”. Universal Journal of Mathematics and Applications 2/3 (September 2019), 107-115. https://doi.org/10.32323/ujma.561120.
JAMA Renukadevi V, Vadakasi S. On Various $g$-Topology in Statistical Metric Spaces. Univ. J. Math. Appl. 2019;2:107–115.
MLA Renukadevi, V. and S. Vadakasi. “On Various $g$-Topology in Statistical Metric Spaces”. Universal Journal of Mathematics and Applications, vol. 2, no. 3, 2019, pp. 107-15, doi:10.32323/ujma.561120.
Vancouver Renukadevi V, Vadakasi S. On Various $g$-Topology in Statistical Metric Spaces. Univ. J. Math. Appl. 2019;2(3):107-15.

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