On Various $g$-Topology in Statistical Metric Spaces

Yıl 2019, , 107 - 115, 30.09.2019

V. Renukadevi S. Vadakasi

https://doi.org/10.32323/ujma.561120

Cited By: 1

Öz

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. 

Anahtar Kelimeler

SM space, type V_D, g-ecart-topology, R-g-topology

Kaynakça

• [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.