【作者】 宋昊；

【导师】 陈东立；

【作者基本信息】 西安建筑科技大学 ， 应用数学， 2011， 硕士

【摘要】 本文首先通过邻域系引出滤子的概念,并对滤子进行了严密分类并讨论了滤子和超滤子的基本性质及其问的关系；其次通过非标准分析理论给出滤单子的定义并研究其性质,利用滤单子给出了滤子和超滤子的非标准刻画；最后研究了一致结构单子的性质,并利用一致空间上的Cauchy滤单子刻画了一致空间的完备性。论文的要点以及主要内容如下第一章中首先简述非标准分析理论,其次介绍了非标准分析理论产生的背景、发展过程以及国内外研究现状。第二章中首先通过拓扑给出了滤子的定义并对滤子进行了分类,其次研究了滤子与超滤子一些性质,最后利用滤子生成一类特殊的连通拓扑。第三章首先讨论了超结构的定义及其相关性质,其次介绍了数理逻辑中的形式语言,以及连接非标准分析与标准分析之间重要的桥梁转换原理,最后讨论了两种经典非标准模型的性质。第四章利用非标准分析理论给出了滤单子的定义,首先讨论了一族滤子的上下确界,其次利用滤单子给出了超滤子的非标准刻画,在此基础上,根据滤子与超滤子间的关系给出了一般滤子的非标准刻画,最后研究了非标准紧化拓扑空间及其商空间的若干性质。第五章应用滤单子理论研究一致空间,首先给出了一致结构的非标准刻画与一致结构单子的性质,其次研究了一致空间上的Cauchy滤单子并利用Cauchy滤单子刻画了一致空间的完备性。更多还原

【Abstract】 In this paper, the concept of a filter is given by neighborhood system and the clas-sification of filters is showed. The concepts of filters and ultrafilters are characterized by nonstandard analysis. The properties of filters and ultrafilters and their relationships are discussed. Further, the monad of a filter is defined and its properties are discussed. Finally, several properties of the monad of a uniformity are discussed and the completeness of uniform spaces is characterized by the monad of Cauchy filters.The main points of this paper are as follows:In the first chapter, the idea of nonstandard analysis is introduced. Then the back-ground, the development and the research states of nonstandard analysis are presented.In the second chapter, the concept and the classification of filter are showed at first. Some properties of filters and ultrafilters are given. Ultimately, a special kind of connection topologies are generated through filters.In the third chapter, the definition and properties of superstructure are discussed. The formal languages in mathematical logic and the transfer principle are presented. Finally, the properties of two kinds of classic nonstandard models are showed.In the fourth chapter, the monad of a filter is defined and the supremum and the infimum of a family of filters are discussed. The concepts of ultrafilters are characterized by the monad of a filter. Based on these concepts, the nonstandard characterizations of filters are showed according to the general relationships between filters and ultrafilters. Finally, some properties of nonstandard compactification of topological spaces and its quotient spaces are studied.In the fifth chapter, uniform spaces are studied by monads of filters. The nonstandard characterizations of uniformity and properties of monads of uniformity are introduced first. Then, monads of Cauchy filters in uniform spaces are studied and the completeness of uniform spaces is characterized by monad of Cauchy filters.更多还原