We present the temporal evolution of magnetic field topology (FT) and its 'bifurcation', i.e. 'magnetic reconnection' taking place in the magnetotail with a southward interplanetary magnetic field. According to the FT theory, the magnetic FT is uniquely determined by the signs of real part of Jacobian matrix eigenvalues of the critical points (CPs) i.e. magnetic null points, if their Jacobian matrix of the CPs is not degenerated. This is because FT can change only at CP locations. The signs of the real part of eigenvalues characterize the magnetic field patterns around CPs. At the CP locations, the magnetic field becomes zero. The CPs can be classified into different types of 'saddles points'. In the saddles points, the eigenvectors span characteristic both one- and two-dimensional manifolds, according to their signs. These manifolds and CPs form FT 'skeletons' and determine essentially the global FT. These FT 'skeletons' that include the CPs, characteristic curves and surfaces which are spanned by the eigenvectors of their CPs provide a clear view of the three-dimensional FT. The change in the skeleton, i.e. the change in the FT, is the so-called 'bifurcation' and allows us to show the occurrence of the magnetic reconnections.