基于内导集运算下的空间刻画

Characterizing Theorem of Space by Internal Derived Operator

拓扑学的理论与方法是近代数学的重要基础,拓扑空间的建立为近代数学的研究打下了坚实的理论基础。对于由邻域、闭包、内部、邻域系、导集运算、边界运算刻画定拓扑的方法已有许多有用的结果,放弃从上述理论构造拓扑空间的方法,转而对点集给出一种新运算——内导集运算的定义,然后利用内导集运算引入拟空间、空间以及拓扑空间概念,进一步研究了拟空间、空间以及拓扑空间的一些相关性质,并且得到与内导集算子可交换的一一映射是同胚映射,从而得到建立的拓扑空间与用开集公理建立的拓扑空间是等价的结论。

Theories and methods of topplogy are an important foundation of modern mathematics.Topological space establishment of the modern mathematics lay a solid theoretical foundation.The method of neighborhood,closure,interior,neighborhood system,derived set operation,the border operation described has many useful topology results.However,this paper abandoned structure from the theoretical approach of topological spaces,but sets a study of the method of Characterizing space in point-set by internal derived operator.First,a new concept of internal derived operator is defined,then the quasi-space,as well as space and topological space,are introduced with internal derived operator.futhermore the related property of quasi-space,as well as space and topological space is studied.Finally,it is proved that one-one mapping which may exchange with the internal derived operator is a homeomorphism,and the conclusion which the topological space to be built and established with the open set axiom is equivalent is obtained.