滤单子理论的研究及其在一致空间中的应用
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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。
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.
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。
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.
引文
[1]A.Robinson. Non-Standard Analysis, Studies in Logic and the Foundations of Mathemat-ics[M]. North-Holland,1966.
[2]M.Davis. Applied nonstandard analysis[M]. New York,1977.
[3]李邦河.非标准分析基础[M].上海科学技术出版社,1986.
[4]W.A.J.Luxemburg. A New Approach to the Theory of Monads[M], ONR Tech. Rept.,1967.
[5]W.A.J.Luxemburg. A general theory of monads Applications of Model Theory to Algebra, Analysis, and Probability[M]. Holt-Rinehart and Winston, New York,1969.
[6]陈东立.关于超实数域*R的基数的一些注记[J].商丘师范学院学报,2002,18(5)：24-26
[7]G.Sack. Saturated Model Theory [M].North-Holland, Amsterdam,1973.
[8]陈东立,马春晖,史艳维.扩大模型的充要条件及其应用[J].西安建筑科技大学学报,2004,36(2)：441-443
[9]潘红霞,陈东立,史艳维.饱和模型的充要条件及其应用[J].纯粹数学与应用数学,2006,22(1)：131-133
[10]A.R.Bernstein. Nonstandard analysis in Studies in Model Theory[J].MAA Studies in Math.,1973,8:35-38
[11]V.Kanovei, S.Shelah. A Definable Nonstandard Model of the Reals[J].Symbolic Logic, 2004,39:159-164.
[12]H.J.Keisler. Ultraproducts Which Are Not Saturated[J]. Journal of Symbolic Logic, 1967,32:23-46
[13]Anderson and Rashid. A Nonstandard Characterization Of Weak Convergence[M]. AMS,1978.
[14]S.Salbany and T.Todorov. Nonstandard Analysis in Point-Set Topology[M]. Vienna,1999.
[15]Gutland. Nonstandard Measure Theory and its Application [J]. Bull. London Math.Soc, 1983,15:529-589
[16]Anderson. Star-finite Representations of Measure Spaces[J]. Math.Soc,1982,271:667-687
[17]F.J.Thayer. Nonstandard Analysis of Graphs[M],2008.
[18]S.Salbany, T.Todorov. Monads and Realcompactness[J].Topology and its Applications. 1994,56:99-104
[19]陈东立.一致结构的单子及其应用[J].纯粹数学与应用数学,2001,17(2)：133-137
[20]P.Hajek. Metamathematics of Fuzzy Logic[M].Dordrecht:Kluwer Academic Publish-ers,1998.
[21]史艳维,陈东立,马春晖.一致可积函数的非标准刻画[J].纯粹数学与应用数学,2008,24(1)：82-83
[22]陈东立,拓扑群上的一致与等度连续的非标准特征及其应用[J],商丘师范学院学报,2002,18(5)：24-26
[23]J.L.Kelly. General Topology[M].New York:Springer-verlag,1975.
[24]H.B.Enderton. Elements of Set Theory[M]. Posts AND Telecom Press,2006.
[25]A.Weil. Sur les espaces a stucture uniforme et sur la topologie generale[M]. Paris, Her-mann,1938.
[26]刘春辉,朱芳芳. Fuzzy蕴涵代数上的一致结构及诱导拓扑[J].山东理工大学学报(自然科学学版),2008,22(5)：56-60
[27]潘丽,陈少白,李文.半一致空间上的最优化问题[J].武汉科技大学学报(自然科学学版)2007,30(6)：666-668
[28]汪义瑞,李生刚.关于有限集上拟一致结构的基数[J].纺织高校基础科学学报,2008,22(1)：45-50
[29]刘德金,刘文虎,王子华.完全正则空间的一个刻画[J].曲阜师范大学学报,2005,31(1)：48-50.
[30]李春玲,李祖泉.邻近拓扑空间和一致拓扑空间[J].佳木斯大学学报(自然科学学版),2001,19(4)：425-426
[31]杨晓伟,徐扬.一致空间中的群结构[J].江西科学,2003,21(1)：78-79
[32]Shi Fugui, Zheng Chongyou,Totally Bounded Pointwise Uniformities and Proximities on Completely Distributive Lattices[J]. Advaces In Mathematics,2001,30(4):322-328
[2]M.Davis. Applied nonstandard analysis[M]. New York,1977.
[3]李邦河.非标准分析基础[M].上海科学技术出版社,1986.
[4]W.A.J.Luxemburg. A New Approach to the Theory of Monads[M], ONR Tech. Rept.,1967.
[5]W.A.J.Luxemburg. A general theory of monads Applications of Model Theory to Algebra, Analysis, and Probability[M]. Holt-Rinehart and Winston, New York,1969.
[6]陈东立.关于超实数域*R的基数的一些注记[J].商丘师范学院学报,2002,18(5)：24-26
[7]G.Sack. Saturated Model Theory [M].North-Holland, Amsterdam,1973.
[8]陈东立,马春晖,史艳维.扩大模型的充要条件及其应用[J].西安建筑科技大学学报,2004,36(2)：441-443
[9]潘红霞,陈东立,史艳维.饱和模型的充要条件及其应用[J].纯粹数学与应用数学,2006,22(1)：131-133
[10]A.R.Bernstein. Nonstandard analysis in Studies in Model Theory[J].MAA Studies in Math.,1973,8:35-38
[11]V.Kanovei, S.Shelah. A Definable Nonstandard Model of the Reals[J].Symbolic Logic, 2004,39:159-164.
[12]H.J.Keisler. Ultraproducts Which Are Not Saturated[J]. Journal of Symbolic Logic, 1967,32:23-46
[13]Anderson and Rashid. A Nonstandard Characterization Of Weak Convergence[M]. AMS,1978.
[14]S.Salbany and T.Todorov. Nonstandard Analysis in Point-Set Topology[M]. Vienna,1999.
[15]Gutland. Nonstandard Measure Theory and its Application [J]. Bull. London Math.Soc, 1983,15:529-589
[16]Anderson. Star-finite Representations of Measure Spaces[J]. Math.Soc,1982,271:667-687
[17]F.J.Thayer. Nonstandard Analysis of Graphs[M],2008.
[18]S.Salbany, T.Todorov. Monads and Realcompactness[J].Topology and its Applications. 1994,56:99-104
[19]陈东立.一致结构的单子及其应用[J].纯粹数学与应用数学,2001,17(2)：133-137
[20]P.Hajek. Metamathematics of Fuzzy Logic[M].Dordrecht:Kluwer Academic Publish-ers,1998.
[21]史艳维,陈东立,马春晖.一致可积函数的非标准刻画[J].纯粹数学与应用数学,2008,24(1)：82-83
[22]陈东立,拓扑群上的一致与等度连续的非标准特征及其应用[J],商丘师范学院学报,2002,18(5)：24-26
[23]J.L.Kelly. General Topology[M].New York:Springer-verlag,1975.
[24]H.B.Enderton. Elements of Set Theory[M]. Posts AND Telecom Press,2006.
[25]A.Weil. Sur les espaces a stucture uniforme et sur la topologie generale[M]. Paris, Her-mann,1938.
[26]刘春辉,朱芳芳. Fuzzy蕴涵代数上的一致结构及诱导拓扑[J].山东理工大学学报(自然科学学版),2008,22(5)：56-60
[27]潘丽,陈少白,李文.半一致空间上的最优化问题[J].武汉科技大学学报(自然科学学版)2007,30(6)：666-668
[28]汪义瑞,李生刚.关于有限集上拟一致结构的基数[J].纺织高校基础科学学报,2008,22(1)：45-50
[29]刘德金,刘文虎,王子华.完全正则空间的一个刻画[J].曲阜师范大学学报,2005,31(1)：48-50.
[30]李春玲,李祖泉.邻近拓扑空间和一致拓扑空间[J].佳木斯大学学报(自然科学学版),2001,19(4)：425-426
[31]杨晓伟,徐扬.一致空间中的群结构[J].江西科学,2003,21(1)：78-79
[32]Shi Fugui, Zheng Chongyou,Totally Bounded Pointwise Uniformities and Proximities on Completely Distributive Lattices[J]. Advaces In Mathematics,2001,30(4):322-328