分形上的拓扑及其性质

Topology of the Fractal in Real Set and Its Properties

鉴于实数集R中的一个s-紧集E在欧氏拓扑下往往是不连通的,利用E本身的欧氏拓扑结构和s-维Hausdorff测度分别给出了E上的使其成为连通紧拓扑空间的2种拓扑的定义,并讨论了2种拓扑的关系及其性质,同时研究了紧拓扑空间上实函数的连续性,为下一步建立分形上函数的微积分理论打下基础.

Because the compact set in real set,E,is often disconnected,in the report,the definition of two kind of topology,which make the fractal connect a compacted topological space,was proposed by means of Euclidean topology and Hausdorff measure,respectively,and the relationship between the two kind of topology and some properties were discussed,the continuity theory for the real function of one variable on fractal was analyzed. The results provide the theory foundation for the further research of fractal analysis.