This paper discusses the evolution of the Fermi surface (FS) topology with doping in electron-doped cuprates within the framework of a one-band Hubbard Hamiltonian, where antiferromagnetism and superconductivity coexist. In the lightly doped insulator, the FS consists of electron pockets around the (π,0) points. The first topological transition (TTI) occurs in the optimally doped region when an additional hole pocket appears at the nodal point. The second topological transition (TTII) occurs in the overdoped regime where antiferromagnetism disappears and a large (π,π)-centered metallic FS is formed. These transitions are supported by Hall effect and penetration depth experiments on Pr2-xCexCuO4-δ (PCCO) and spectroscopic measurements on Nd2-xCexCuO4-δ (NCCO). The FS topology is reflected in the temperature dependence of the penetration depth λ. Below TTI, λ behaves like an 's'-wave (nodeless d-wave) form, transitioning to a mixed d+s'-wave form above TTI, and to a pure d-wave form above TTII. The superfluid density ns varies exponentially with temperature in the underdoped regime due to strong AFM correlations, but becomes linear-in-T near optimal doping. The doping evolution of ns provides insight into the FS topology. Hall effect measurements show that the Hall coefficient changes from negative to positive near optimal doping, indicating a change in carrier type from electron-like to hole-like. The Hall data at intermediate doping can be fitted by a weighted average of low and high doping data, suggesting two-carrier conduction. This occurs at the same doping where nodal pockets first appear (TTI). The coexistence of AFM and SC orders leads to non-Fermi-liquid behavior and possible Bose-Einstein condensation. The model of two topological transitions in the FS is consistent with experimental results and provides a framework for understanding the doping evolution of various properties in electron-doped cuprates.This paper discusses the evolution of the Fermi surface (FS) topology with doping in electron-doped cuprates within the framework of a one-band Hubbard Hamiltonian, where antiferromagnetism and superconductivity coexist. In the lightly doped insulator, the FS consists of electron pockets around the (π,0) points. The first topological transition (TTI) occurs in the optimally doped region when an additional hole pocket appears at the nodal point. The second topological transition (TTII) occurs in the overdoped regime where antiferromagnetism disappears and a large (π,π)-centered metallic FS is formed. These transitions are supported by Hall effect and penetration depth experiments on Pr2-xCexCuO4-δ (PCCO) and spectroscopic measurements on Nd2-xCexCuO4-δ (NCCO). The FS topology is reflected in the temperature dependence of the penetration depth λ. Below TTI, λ behaves like an 's'-wave (nodeless d-wave) form, transitioning to a mixed d+s'-wave form above TTI, and to a pure d-wave form above TTII. The superfluid density ns varies exponentially with temperature in the underdoped regime due to strong AFM correlations, but becomes linear-in-T near optimal doping. The doping evolution of ns provides insight into the FS topology. Hall effect measurements show that the Hall coefficient changes from negative to positive near optimal doping, indicating a change in carrier type from electron-like to hole-like. The Hall data at intermediate doping can be fitted by a weighted average of low and high doping data, suggesting two-carrier conduction. This occurs at the same doping where nodal pockets first appear (TTI). The coexistence of AFM and SC orders leads to non-Fermi-liquid behavior and possible Bose-Einstein condensation. The model of two topological transitions in the FS is consistent with experimental results and provides a framework for understanding the doping evolution of various properties in electron-doped cuprates.