Monotone open homogeneity of Sierpiński curve

This paper answers positively the open question of whether or not the Sierpiński curve is homogeneous with respect to monotone open maps. It constructs a monotone open map from the Sierpiński curve onto the Sierpiński curve. The map takes a boundary point of a complementary region onto a point which is not a boundary point of a complementary region and vice versa. We construct the map by building a continuous decomposition of the Sierpiński curve so that the decomposition space is homeomorphic to the Sierpiński curve. Each decomposition element is a nondegenerate cellular continuum except for one which is a simple closed curve: the boundary of a complementary region.