强模糊范数所诱导的模糊化拓扑
发布时间：2019-05-07 11:50
【摘要】：模糊赋范线性空间理论是模糊分析学的重要组成部分,模糊赋范空间中的拓扑结构一直成为众多学者的关注热点.基于强模糊度量空间,本文对强模糊赋范空间中的模糊化拓扑进行了较为细致的研究,主要内容如下:一、引入强模糊赋范线性空间的概念,给出强模糊赋范空间上具有层次结构的分明拓扑族,并利用此分明拓扑族确定强模糊赋范空间上的模糊化拓扑.证明了模糊化拓扑与线性运算的相容性.给出了强模糊赋范线性空间序列收敛度的模糊范数式刻画,利用序列收敛度的概念,给出了模糊化拓扑与线性运算相容性的另一种证明.研究了强模糊赋范线性空间中映射的连续性问题.二、给出了有限个模糊赋范线性空间乘积模糊范数的两种定义,证明了两种模糊范数确定的拓扑等价.引入了可数个模糊赋范线性空间乘积的定义,研究了可数个模糊赋范乘积空间的拓扑结构.特别地,证明了有限个强模糊赋范乘积空间确定的模糊化拓扑等于其因子空间模糊化拓扑乘积,可数个强模糊赋范乘积空间确定的模糊化拓扑强于其因子空间模糊化拓扑乘积.
[Abstract]:The theory of fuzzy normed linear space is an important part of fuzzy analysis. The topological structure of fuzzy normed space has always been the focus of many scholars. Based on strong fuzzy metric space, the fuzzy topology in strongly fuzzy normed space is studied in detail. The main contents are as follows: first, the concept of strongly fuzzy normed linear space is introduced. An explicit topological family with hierarchical structure on strongly fuzzy normed spaces is given, and the fuzzy topology on strongly fuzzy normed spaces is determined by using this distinct topological family. The compatibility between fuzzy topology and linear operation is proved. In this paper, the fuzzy norm formula of convergence degree of strongly fuzzy normed linear space is given. By using the concept of convergence degree of sequence, another proof of compatibility between fuzzy topology and linear operation is given. The continuity of mappings in strongly fuzzy normed linear spaces is studied. Secondly, two definitions of the product fuzzy norm of finite fuzzy normed linear spaces are given, and the topological equivalence of the two fuzzy norms is proved. In this paper, the definition of product of countable fuzzy normed linear spaces is introduced, and the topological structure of countable fuzzy normed product spaces is studied. In particular, it is proved that the fuzzy topology of finite strong fuzzy normed product spaces is equal to the fuzzy topological product of its factor space, and the fuzzy topology of countable strong fuzzy normed product space is stronger than the fuzzy topological product of its factor space.
【学位授予单位】：南京师范大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O177
[Abstract]:The theory of fuzzy normed linear space is an important part of fuzzy analysis. The topological structure of fuzzy normed space has always been the focus of many scholars. Based on strong fuzzy metric space, the fuzzy topology in strongly fuzzy normed space is studied in detail. The main contents are as follows: first, the concept of strongly fuzzy normed linear space is introduced. An explicit topological family with hierarchical structure on strongly fuzzy normed spaces is given, and the fuzzy topology on strongly fuzzy normed spaces is determined by using this distinct topological family. The compatibility between fuzzy topology and linear operation is proved. In this paper, the fuzzy norm formula of convergence degree of strongly fuzzy normed linear space is given. By using the concept of convergence degree of sequence, another proof of compatibility between fuzzy topology and linear operation is given. The continuity of mappings in strongly fuzzy normed linear spaces is studied. Secondly, two definitions of the product fuzzy norm of finite fuzzy normed linear spaces are given, and the topological equivalence of the two fuzzy norms is proved. In this paper, the definition of product of countable fuzzy normed linear spaces is introduced, and the topological structure of countable fuzzy normed product spaces is studied. In particular, it is proved that the fuzzy topology of finite strong fuzzy normed product spaces is equal to the fuzzy topological product of its factor space, and the fuzzy topology of countable strong fuzzy normed product space is stronger than the fuzzy topological product of its factor space.
【学位授予单位】：南京师范大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O177
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