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… Click to show full abstract
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.
