On Various $g$-Topology in Statistical Metric Spaces

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