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Topological phase transitions in a three-dimensional (3D) topological insulator (TI) with an exchange field of strength g are studied by calculating spin Chern numbers C ^± ( k _z ) with momentum k _z as a parameter. When | g | exceeds a critical value g _C , a transition of the 3D TI into a Weyl semimetal occurs, where two Weyl points appear as critical points separating k _z regions with different first Chern numbers. For | g | <  g _C , C ^± ( k _z ) undergo a transition from ±1 to 0 with increasing | k _z | to a critical value k ^C _z . Correspondingly, surface states exist for | k _z | <  k ^C _z , and vanish for | k _z | ⩾  k ^C _z . The transition at | k _z | =  k ^C _z is accompanied by closing of the bulk spin spectrum gap rather than the energy gap.