[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.
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.
[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.
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.