  We report a fine tuned doping study of strongly overdoped
Bi$_2$Sr$_2$CaCu$_2$O$_{8+\delta}$ single crystals using electronic Raman
scattering. Combined with theoretical calculations, we show that the doping, at
which the normal state pseudogap closes, coincides with a Lifshitz quantum
phase transition where the active hole-like Fermi surface becomes
electron-like. This conclusion suggests that the microscopic cause of the
pseudogap is sensitive to the Fermi surface topology. Furthermore, we find that
the superconducting transition temperature is unaffected by this transition,
demonstrating that their origins are different on the overdoped side.
