The Metrization Problem in [0,1]-Topology

Abstract

This paper discusses the classification of fuzzy metrics based on their continuity conditions, dividing them into Erceg, Deng, Shi, and Chen metrics. It explores the relationships between these types of fuzzy metrics, concluding that a Deng metric in [0,1]-topology must also be Erceg, Chen, and Shi metrics. This paper also proves that the product of countably many Deng pseudo-metric spaces remains a Deng pseudo-metric space, and demonstrates some σ-locally finite properties of Deng metric space. Additionally, this paper constructs two interrelated mappings based on normal space and concludes that, if a [0,1]-topological space is T1 and regular, and its topology has a σ-locally finite base, then it is Deng-metrizable, and thus Erceg-, Shi-, and Chen-metrizable as well.

Keywords

• fuzzy point
• [0,1]-topology
• Deng pseudo-metric
• <i>σ</i>-locally finite base
• regular
• <i>T</i>₁-space