Existence of differentiable connections on top spaces

In this paper, differentiable connections on top spaces are studied and some conditions on which there is no differentiable connection passing from a given point in the top space are found. In a special case, the Euclidean space ???? is considered as a top space and the existence of differentiable connections is studied. Finally, we prove that the smoothness condition of the inverse map in the definition of a top space is redundant.