Characterizations of Pointwise Pseudometrics via Pointwise Closed-Ball Systems

作者：Chong Shen, Yi Shi, Fu-Gui Shi, Hadrian Andradi · 发表于：IEEE Transactions on Fuzzy Systems · 年份：2021 · DOI：10.1109/tfuzz.2021.3054466 · 被引用次数：6 · 研究领域：Fuzzy and Soft Set Theory、Fixed Point Theorems Analysis、Advanced Algebra and Logic

Pointwise pseudoquasi-metrics play an important role in the theory of lattice-valued topology ($L$-topology). Bearing in mind that closed balls and their relations with pseudoquasi-metrics have historically attracted the attention of mathematicians, it is very surprising that no attention has been paid to the relations between pointwise pseudoquasi-metrics and closed balls. In this article, we first introduce the concept of pointwise closed-ball systems and prove that the resulting category is isomorphic to that of pointwise pseudoquasi-metrics. Subsequently, we study the topological properties of pointwise pseudoquasi-metrics via pointwise closed-ball systems. Interestingly, the$L$-topologies defined by open sets and complements of closed sets coincide for any pointwise pseudometric. Finally, we expose an important theoretical application of the pointwise closed-systems in providing a different and relatively simpler proof of the celebrated metrization theorem of the$L$-fuzzy real line.