  We introduce a new topology, weaker than the gap topology, on the space of
selfadjoint operators affiliated to a semifinite von Neumann algebra. We define
the real-valued spectral flow for a continuous path of selfadjoint
Breuer-Fredholm operators in terms of a generalization of the winding number.
We compare our definition with Phillips' analytical definition and derive
integral formulas for the spectral flow for certain paths of unbounded
operators with common domain, generalizing those of Carey--Phillips.
Furthermore we prove the homotopy invariance of the real-valued index. As an
example we consider invariant symmetric elliptic differential operators on
Galois coverings.
