Sierpinski垫的Whitney临界集

以Sierpinski垫为例，进一步研究了不是Whitney临界集的分形集可以包含Whitney临界集的问题。首先，在Sierpinski垫中构造一个连通集合E，E是由9个压缩比为1／8的压缩函数生成相似集且满足开集条件，它的Hausdorff维数为ln9/ln8;其次，在连通集合E上的构造一个可微函数，利用该函数分3种情形证明了E是一个Whitney临界集，于是得到不是Whitney临界集的Sierpinski垫可以包含Whitney临界集E。

Whitney sets on Sierpinski gasket

Taking the Sierpinski gasket as an example,the paper studied the problem of Fractal set that is not Whitney′s critical set could contain Whitney′s critical set.Firstly,a connected set E was constructed on the Sierpinski gasket which was self-similar set resulted from nine contraction function with contraction ratio of 1/8 and satisfied the open set conditions, and whose Hausdorff dimension was ln9/ln8;secondly,a differential function was constructed on the connected set E to prove that E was a Whitney′s critical set by dividing the differential function into three cases.Thus reaching the conclusion that the Sierpinski gasket which was not Whitney′s critical set could contain Whitney′s critical set E.