We study the topology of the Julia J set of a quadratic Cremer polynomial P. It is known that such a Julia set is a non-locally connected, non-separating one dimensional plane continuum. We show that if there exists an external angle whose impression does not contain the fixed Cremer point p, then J is connected im kleinen at a dense set of points and these points are contained in a unique, degenerate impression. (authors-A.Blokh and L.Oversteegen)