模糊k-度量的新定义及其在拓扑和凸结构理论中的应用

Fuzzy k-metric, a generalization of both classic k-metric and fuzzy metric, is an emerging research direction in the field of fuzzy metric. Combining the relevant knowledge of fuzzy topology, fuzzy convergence and fuzzy convex structure, this project intends to do the following three research work: (1)In order to show the nature of fuzzy mathematics, we plan to give a new kind definition of fuzzy k-(pseudo)metric, to establish different kinds of fuzzifying structures induced by a fuzzy k-pseudometric, and to consider the convergence degree of sequences, the degree of Cauchy sequences, the degree of compactness, the degree of totally boundedness and the degree of completeness in a fuzzy k-metric space. (2)In order to enriched the research contents of fuzzy k-metric spaces, we plan to discuss the relationship between fuzzy k-pseudometrics and classic k-pseudometrics, and to establish the characteristic representation theorem of fuzzy k-pseudometrics. Besides, we also plan to analyze the equivalent characterizations of degress of convergence, compactness, completeness etc., and to explore the relationship between the fuzzifying structures indirectly induced by a family of classic k-pseudometrics and the fuzzifying structures directly induced by a fuzzy k-pseudometric. (3)So as to promote the fusion of fuzzy k-metric and fuzzy convex structure, we intend to establish a fuzzifying geodesic k-convex structure on a fuzzy k-metric space, and to discuss the basic framework of fuzzifying geodesic k-convex structure in detail. Therefore, this project wihch may provide a new research tool in the application fields of fuzzy metrics and fuzzy convex structures, has important theoretical and practical value.

模糊k-度量是经典k-度量和模糊度量的双重推广，是模糊度量领域一个新兴的研究方向。结合模糊拓扑、模糊收敛、模糊凸结构的相关知识，本项目拟考虑如下三方面的工作：(1)给出模糊k-(伪)度量的新定义，建立模糊k-伪度量诱导的多种模糊化结构，考虑模糊k-度量空间中序列的收敛度、柯西序列度、紧度、完全有界度、完备度等内容，充分展现模糊数学程度化的思想；(2)分析模糊k-伪度量和经典k-伪度量的关系，建立模糊k-伪度量的特征族表示定理，讨论收敛度、紧度、完备度等的特征族刻画及探讨特征族间接诱导的模糊化结构和直接诱导的模糊化结构的关系，丰富模糊k-度量空间的研究内容；(3)建立模糊k-度量空间上的模糊化测地k-凸结构及其基本研究框架，促进模糊k-度量和模糊凸结构的融合。本项目的研究内容能为模糊度量和模糊凸结构理论在应用学科领域提供一种新的研究工具，具有重要的理论意义和实际价值。

模糊k-伪度量是经典k-伪度量和模糊伪度量的双重模糊推广，本项目主要介绍模糊k-伪度量的新定义及讨论模糊k-伪度量在模糊拓扑和模糊凸结构中的应用。首先，给出了符合经典k-伪度量语义推广的模糊k-伪度量的新定义。在此基础上，建立了由模糊k-伪度量诱导的模糊化拓扑、模糊化一致结构、模糊化领域系统、模糊化闭包算子、模糊化内部算子等模糊化结构。其次，讨论了模糊k-伪度量和经典k-伪度量的关系，证明了模糊k-伪度量全体和满足一定条件的经典k-伪度量全体之间存在着一一对应关系。除此之外，建立了模糊k-伪度量的特征族表示定理，并得到了由特征族表示定理间接诱导的模糊化结构和由模糊k-伪度量直接诱导的模糊化结构保持一致，丰富了模糊k-伪度量空间的研究内容。最后，构造了模糊k-伪度量上的模糊化测地区间算子、模糊化凸包算子的具体数学表达式。并且，建立了模糊k-伪度量空间上的模糊化测地k-凸结构及其基本研究框架，促使模糊k-伪度量理论和模糊凸结构理论的相互融合。本项目的研究内容具有重要的理论和实际价值。