Expansion Theory of Deng’s Metric in [0,1]-Topology

The aim of this paper is to focus on a fuzzy metric called Deng’s metric in [0,1]-topology. Firstly, we will extend the domain of this metric function from M0×M0 to M×M, where M0 and M are defined as the sets of all special fuzzy points and all standard fuzzy points, respectively. Secondly, we will further extend this metric to the completely distributive lattice LX and, based on this extension result, we will compare this metric with the other two fuzzy metrics: Erceg’s metric and Yang-Shi’s metric, and then reveal some of its interesting properties, particularly including its quotient space. Thirdly, we will investigate the relationship between Deng’s metric and Yang-Shi’s metric and prove that a Deng’s metric must be a Yang-Shi’s metric on IX, and consequently an Erceg’s metric. Finally, we will show that a Deng’s metric on IX must be Q−C1, and Deng’s metric topology and its uniform structure are Erceg’s metric topology and Hutton’s uniform structure, respectively.