  The axial anomaly of Ginsparg-Wilson fermion operator $D$ is discussed in
general for the operator $R$ which enters the chiral symmetry breaking part in
the Ginsparg-Wilson relation. The axial anomaly and the index of $D$ as well as
the exact realization of the Atiyah-Singer index theorem on the lattice are
determined solely by the topological characteristics of the chirally symmetric
operator $D_c$ in the chiral limit $ R \to 0 $.
