Whitneyʹs critical set in fractal

The problem is concerned about how large (e.g. the Hausdorff dimension) is Whitneyʹs critical set contained in a given fractal. For this, we prove that the Moran arc, an arc containing a Moran set, is a Whitneyʹs critical set. The excellent open set condition is defined, when the condition holds, the associated self-similar set contains a Whitneyʹs critical subset of full dimension. As its application, the Sierpinski gasket and Koch curve have Whitneyʹs critical subset of full dimension. Finally, we provide a self-similar tree which never contains any Whitneyʹs critical set.