On Various $g$-Topology in Statistical Metric Spaces

Authors : V. Renukadevi, S. Vadakasi

Pages : 107-115

Doi:10.32323/ujma.561120

View : 9 | Download : 6

Publication Date : 2019-09-30

Article Type : Research

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. 
Keywords : SM space, type V_D, g-ecart-topology, R-g-topology