拓朴群上的超拓扑群
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摘要
随着FUZZY数学基础的研究，突出了集值映射的重要性，于是引出
了各种结构的提升问题，例如序结构、拓扑结构、可测结构以及代数结
构等等。文[1]考虑了代数结构的提升问题，首次提出了HX群的概念。
文[2][3]分别定义了正规幂群、一致幂群，较系统地研究了各种幂群的结
构。文[4]—[9]分别研究了各种幂群的性质、结构、分类、同态和同构关
系。文[10][11]将拓扑群的两个数学结构分别向幂集上提升，得到了超拓
扑群，对拓扑群的提升做了突破性的工作。
本文在文[10][11]的基础上，进一步研究了超拓扑群，首先以邻域定
义拓扑，继而得出了超拓扑群的定义，然后证明了超拓扑群的正则性和
齐性，又定义了超拓扑群的子群、正规子群与商群，同时也研究了它们
的一些基本性质，最后研究了超拓扑群的同胚、同态、同构及连通性。
With the study of Fuzzy mathematics, the importance of Set value
mapping has been highlighted. Then the upgrade of all kings of the structures,
such as ordered, topological, measurable structure and algebraic structure,
etc, has been brought about. Paper [1] considered the upgrade of algebraic
structure, and the concept of FIX group was raised firstly. Paper [2] [3) not
only defined the concept of normal FIX group and uniform HX group, but
also studied the structures of all kinds of HX group systematic. The
properties, structures, classifications, homomorphism and isomorphism of all
kinds of FIX group have been studies respective in [4]9]. In the paper [101
two kinds of mathematical structures of topological group is upgraded, and
the concept of hyper opological group is raised, and some breakthrough is
done for the upgrade of topological group.
On the basis of paper [l0][l1), this paper gives some further description
for FIyper opological group, proved the uniform and regular properties of
the fIX group of topological group, defined the subgroup, normal subgroup,
quotient group of the HX group of topological group, and in the meantime,
studied some properties of theirs. Finally it homemorphism,
homomorphism, isomorphism and connectivity will be considered.
了各种结构的提升问题，例如序结构、拓扑结构、可测结构以及代数结
构等等。文[1]考虑了代数结构的提升问题，首次提出了HX群的概念。
文[2][3]分别定义了正规幂群、一致幂群，较系统地研究了各种幂群的结
构。文[4]—[9]分别研究了各种幂群的性质、结构、分类、同态和同构关
系。文[10][11]将拓扑群的两个数学结构分别向幂集上提升，得到了超拓
扑群，对拓扑群的提升做了突破性的工作。
本文在文[10][11]的基础上，进一步研究了超拓扑群，首先以邻域定
义拓扑，继而得出了超拓扑群的定义，然后证明了超拓扑群的正则性和
齐性，又定义了超拓扑群的子群、正规子群与商群，同时也研究了它们
的一些基本性质，最后研究了超拓扑群的同胚、同态、同构及连通性。
With the study of Fuzzy mathematics, the importance of Set value
mapping has been highlighted. Then the upgrade of all kings of the structures,
such as ordered, topological, measurable structure and algebraic structure,
etc, has been brought about. Paper [1] considered the upgrade of algebraic
structure, and the concept of FIX group was raised firstly. Paper [2] [3) not
only defined the concept of normal FIX group and uniform HX group, but
also studied the structures of all kinds of HX group systematic. The
properties, structures, classifications, homomorphism and isomorphism of all
kinds of FIX group have been studies respective in [4]9]. In the paper [101
two kinds of mathematical structures of topological group is upgraded, and
the concept of hyper opological group is raised, and some breakthrough is
done for the upgrade of topological group.
On the basis of paper [l0][l1), this paper gives some further description
for FIyper opological group, proved the uniform and regular properties of
the fIX group of topological group, defined the subgroup, normal subgroup,
quotient group of the HX group of topological group, and in the meantime,
studied some properties of theirs. Finally it homemorphism,
homomorphism, isomorphism and connectivity will be considered.
引文
[1] Li Hongxing, Duen Qinzhi, Wang Peizhuang , Hypergroups, BUSEFAL(法刊).23(1985)
[2] 张振良，李洪兴，汪培庄，正规超群与商群的关系。数学季刊3(1987) 42-45
[3] 钟育彬，幂群的结构与相互关系。数学季刊4(1990) 102-106
[4] Zhang Zhenliang, The Properties of HX Groups. Italian Journal of Pure and Applied Mathematics (意刊).No.2(1997) 97-106
[5] Zhang Zhenliang, Classifications of HX Groups and Their chair of normal subgroup, Italian Journal of Pure and Applied Mathematics(意刊) (1999)
[6] 张振良，幂群的结构与分类。纯粹数学与应用数学1(1998) 36-41
[7] 张振良，幂群的结构与同态、同构关系。昆明理工大学学报1(1998) 92-99
[8] 李洪兴，正则HX群的同态与同构。模糊系统与数学1(1990) 1-7
[9] 李洪兴，汪培庄，幂群，应用数学。2(1988) 1-4
[10] 罗承忠，钟育彬, 拓扑群的提升。北京师范大学学报(自然科学 版)3(1993) 61-63
[11] 李洪兴，段钦治，基于幂群的超拓扑群。北京师范大学学报(自 然科学版)1(1996) 7-9
[12] 邦德列雅金，连续群(上册)。科学出版社1978
[13] 黎景辉，冯绪宁，拓扑群引论。科学出版社1991
[14] 张振良，正规幂环与一致幂环。纯粹数学与应用数1(2001)
[15] Zhang Zhenliang, Structures and Classifications of HX Ring. Italian Journal of Pure and Aplied Mathematics (意刊) (2001)
[15] 杨文泽，以群的非空子集为元素的群。西南师范学院学报2(1985) ，106-108
[16] 杨文泽，幂群与它的生成群。数学实践与认识4(1998) 345-349
[17] 罗承忠，米洪海，FUZZY幂群。模糊系统与数学1(1994) 1-9
[18] 钟育彬，FUZZY拓扑群的基数定理。模糊系统与数学2(1998) 61-69
[19] 汪培庄，李洪兴著，模糊系统理论与模糊计算机。科学出版社1996
[20] 罗承忠，模糊集引论。北京师范大学出版社1989
[21] 熊金城编，点集拓扑讲议。高等教育出版社1998
[22] 汪培庄著，模糊集与随机集落影。北京师范大学出版社1985
[23] Zheng Zhenliang, Antitonic Set-Valued Mappings and Fuzzy Sets. BUSEFAL.9(1989)
[24] 张文修著，模糊数学基础。西安交通大学出版社1984
[25] 刘应明 主编，刘旺金，何家儒著，模糊数学导论。四川教育出版 社1992
[26] 张振良，集合套的落影函数。模糊系统与数学课(1990) 38-44
[27] 刘文奇，罗承忠，集合环。模糊系统与数学课(1997) 34-39
[28] 李洪兴，汪培庄编著，模糊数学。国防工业出版社1994
[29] 罗承忠，扩展原理和FUZZY数学。模糊数学系984) 109-116
[30] 罗承忠，扩展原理和FUZZY数学(Ⅱ)模糊数学系984) 105-114
[31] 李洪兴，从模糊控制的数学本看模糊逻辑的成功。模糊系统与数 学系(1995) 1-13
[32] 熊金编著，近世代数。武汉大学出版社 1984
[33] 吴品三编，近世代数。人民教育出版社 1981
[2] 张振良，李洪兴，汪培庄，正规超群与商群的关系。数学季刊3(1987) 42-45
[3] 钟育彬，幂群的结构与相互关系。数学季刊4(1990) 102-106
[4] Zhang Zhenliang, The Properties of HX Groups. Italian Journal of Pure and Applied Mathematics (意刊).No.2(1997) 97-106
[5] Zhang Zhenliang, Classifications of HX Groups and Their chair of normal subgroup, Italian Journal of Pure and Applied Mathematics(意刊) (1999)
[6] 张振良，幂群的结构与分类。纯粹数学与应用数学1(1998) 36-41
[7] 张振良，幂群的结构与同态、同构关系。昆明理工大学学报1(1998) 92-99
[8] 李洪兴，正则HX群的同态与同构。模糊系统与数学1(1990) 1-7
[9] 李洪兴，汪培庄，幂群，应用数学。2(1988) 1-4
[10] 罗承忠，钟育彬, 拓扑群的提升。北京师范大学学报(自然科学 版)3(1993) 61-63
[11] 李洪兴，段钦治，基于幂群的超拓扑群。北京师范大学学报(自 然科学版)1(1996) 7-9
[12] 邦德列雅金，连续群(上册)。科学出版社1978
[13] 黎景辉，冯绪宁，拓扑群引论。科学出版社1991
[14] 张振良，正规幂环与一致幂环。纯粹数学与应用数1(2001)
[15] Zhang Zhenliang, Structures and Classifications of HX Ring. Italian Journal of Pure and Aplied Mathematics (意刊) (2001)
[15] 杨文泽，以群的非空子集为元素的群。西南师范学院学报2(1985) ，106-108
[16] 杨文泽，幂群与它的生成群。数学实践与认识4(1998) 345-349
[17] 罗承忠，米洪海，FUZZY幂群。模糊系统与数学1(1994) 1-9
[18] 钟育彬，FUZZY拓扑群的基数定理。模糊系统与数学2(1998) 61-69
[19] 汪培庄，李洪兴著，模糊系统理论与模糊计算机。科学出版社1996
[20] 罗承忠，模糊集引论。北京师范大学出版社1989
[21] 熊金城编，点集拓扑讲议。高等教育出版社1998
[22] 汪培庄著，模糊集与随机集落影。北京师范大学出版社1985
[23] Zheng Zhenliang, Antitonic Set-Valued Mappings and Fuzzy Sets. BUSEFAL.9(1989)
[24] 张文修著，模糊数学基础。西安交通大学出版社1984
[25] 刘应明 主编，刘旺金，何家儒著，模糊数学导论。四川教育出版 社1992
[26] 张振良，集合套的落影函数。模糊系统与数学课(1990) 38-44
[27] 刘文奇，罗承忠，集合环。模糊系统与数学课(1997) 34-39
[28] 李洪兴，汪培庄编著，模糊数学。国防工业出版社1994
[29] 罗承忠，扩展原理和FUZZY数学。模糊数学系984) 109-116
[30] 罗承忠，扩展原理和FUZZY数学(Ⅱ)模糊数学系984) 105-114
[31] 李洪兴，从模糊控制的数学本看模糊逻辑的成功。模糊系统与数 学系(1995) 1-13
[32] 熊金编著，近世代数。武汉大学出版社 1984
[33] 吴品三编，近世代数。人民教育出版社 1981