Alexandrov空間的公理體系
發(fā)布時間：2018-11-05 08:49
【摘要】：Alexandrov空間是指開集族對任意并和任意交都封閉的拓撲空間,由前蘇聯數學家Alexandrov于1937年提出。可能是由于這一類空間的公理體系比較對稱和簡單,上世紀九十年代前很少有關于這方面的專門研究;九十年代后由于和數字拓撲中的有限拓撲空間有很多相似的性質,Alexandrov空間在數字拓撲學領域得到了廣泛的應用。進入二十一世紀后,Alexandrov空間的研究出現在粗糙集理論中,這是因為Pawlak粗糙集對應的拓撲空間恰是Alexandrov空間,并且等價關系和滿足對稱分離性的Alexandrov空間之間具有一一對應關系。本文從鄰域系統(tǒng)、閉包算子、內部算子、導算子及其一些衍生算子、二元關系、特殊化序和無點化序等方面對Alexandrov空間的公理體系進行了系統(tǒng)整理和總結性研究,從范疇論的角度建立了Alexandrov空間和鄰域系統(tǒng)、閉包算子空間、內部算子空間、導算子空間、預序集和完備生成格等之間的范疇同構。上述結果表明,Alexandrov空間除了具有拓撲特征之外,還具有明顯的二元關系特征、序特征和代數特征,將在基于關系的聚類分析理論、粗糙集理論和數據挖掘等領域中具有重要應用。
[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.
【學位授予單位】：河北科技大學
【學位級別】：碩士
【學位授予年份】：2017
【分類號】：O189.11
本文編號：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.
【學位授予單位】：河北科技大學
【學位級別】：碩士
【學位授予年份】：2017
【分類號】：O189.11
【參考文獻】
相關期刊論文 前1條
1 楊小飛;李生剛;;預導算子、預差導算子及預拓撲[J];山東大學學報(理學版);2007年12期
，本文編號：2311558
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