Alexandrov空间的公理体系
发布时间：2018-11-05 08:49
【摘要】：Alexandrov空间是指开集族对任意并和任意交都封闭的拓扑空间,由前苏联数学家Alexandrov于1937年提出。可能是由于这一类空间的公理体系比较对称和简单,上世纪九十年代前很少有关于这方面的专门研究;九十年代后由于和数字拓扑中的有限拓扑空间有很多相似的性质,Alexandrov空间在数字拓扑学领域得到了广泛的应用。进入二十一世纪后,Alexandrov空间的研究出现在粗糙集理论中,这是因为Pawlak粗糙集对应的拓扑空间恰是Alexandrov空间,并且等价关系和满足对称分离性的Alexandrov空间之间具有一一对应关系。本文从邻域系统、闭包算子、内部算子、导算子及其一些衍生算子、二元关系、特殊化序和无点化序等方面对Alexandrov空间的公理体系进行了系统整理和总结性研究,从范畴论的角度建立了Alexandrov空间和邻域系统、闭包算子空间、内部算子空间、导算子空间、预序集和完备生成格等之间的范畴同构。上述结果表明,Alexandrov空间除了具有拓扑特征之外,还具有明显的二元关系特征、序特征和代数特征,将在基于关系的聚类分析理论、粗糙集理论和数据挖掘等领域中具有重要应用。
[Abstract]:Alexandrov space is a topological space in which the open set family is closed to any and any intersection. It was proposed by the former Soviet mathematician Alexandrov in 1937. It may be that the axiomatic system of this kind of space is relatively symmetrical and simple, and there were few special studies on this aspect before the 1990s; Since the 1990s, Alexandrov spaces have been widely used in the field of digital topology because of their similar properties to the finite topological spaces in digital topology. In the 21 century, the study of Alexandrov space appears in rough set theory, because the topological space corresponding to Pawlak rough set is exactly Alexandrov space, and there is a one-to-one correspondence between the equivalence relation and the Alexandrov space which satisfies the symmetric separation. In this paper, the axioms of Alexandrov spaces are systematically arranged and summarized from the aspects of neighborhood system, closure operator, interior operator, derivative operator and some derivation operators, binary relations, specialization order and non-point order, etc. From the perspective of category theory, the category isomorphism between Alexandrov space and neighborhood system, closure operator space, interior operator space, derivative operator space, preordered set and complete generating lattice is established. The above results show that the Alexandrov space not only has topological features, but also has obvious binary relation features, order features and algebraic features, which will be based on the theory of relational clustering analysis. Rough set theory and data mining have important applications.
【学位授予单位】：河北科技大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O189.11
本文编号：2311558
[Abstract]:Alexandrov space is a topological space in which the open set family is closed to any and any intersection. It was proposed by the former Soviet mathematician Alexandrov in 1937. It may be that the axiomatic system of this kind of space is relatively symmetrical and simple, and there were few special studies on this aspect before the 1990s; Since the 1990s, Alexandrov spaces have been widely used in the field of digital topology because of their similar properties to the finite topological spaces in digital topology. In the 21 century, the study of Alexandrov space appears in rough set theory, because the topological space corresponding to Pawlak rough set is exactly Alexandrov space, and there is a one-to-one correspondence between the equivalence relation and the Alexandrov space which satisfies the symmetric separation. In this paper, the axioms of Alexandrov spaces are systematically arranged and summarized from the aspects of neighborhood system, closure operator, interior operator, derivative operator and some derivation operators, binary relations, specialization order and non-point order, etc. From the perspective of category theory, the category isomorphism between Alexandrov space and neighborhood system, closure operator space, interior operator space, derivative operator space, preordered set and complete generating lattice is established. The above results show that the Alexandrov space not only has topological features, but also has obvious binary relation features, order features and algebraic features, which will be based on the theory of relational clustering analysis. Rough set theory and data mining have important applications.
【学位授予单位】：河北科技大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O189.11
【参考文献】
相关期刊论文 前1条
1 杨小飞;李生刚;;预导算子、预差导算子及预拓扑[J];山东大学学报(理学版);2007年12期
，本文编号：2311558
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