n维欧氏空间一类特殊集合的维数和势

Dimension and potential of a special set in N-dimensional Euclidean space

首先将闭区间[0,1]上的Cantor集的构造方法推广到n维欧氏空间,得到n-维Cantor集.通过维数的计算公式,得到n-维Cantor集维数.然后,从Cantor集的性质出发,证明了n-维Cantor集仍然具有连续统的势.最后,对n-维Cantor集作进一步扩展,获得了n-维扩展Cantor集的维数和势的相应结果.

Firstly,this paper promotes the method of creating Cantor Cluster in the closed interval to the n-dimension Euclidean space.By means of calculation formula of dimensions,this paper obtains the dimension of n-D Cantor Cluster,and,based on the nature of Cantor Cluster,proves that the n-D Cantor Cluster still has continuous potential.Finally,this paper further promotes the n-D Cantor Cluster and obtains the dimension as well as potential of the promoted n-D Cantor Cluster.