The electronic structures, magnetism, and half-metallicity of the full-Heusler $Co_2TiSn$(001) surfaces have been investigated by using the all-electron full-potential linearized augmented plane wave method within the generalized gradient approximation. We have considered both of the Co atoms terminated(Co-term) and the TiSn atoms terminated(TiSn-term) surfaces. From the calculated density of states, we found that the half-metallicity was destroyed at the surface of the Co-term, while the half-metallicity was retained at the TiSn-term. For the surface of the Co-term, due to the reduced coordination number the occupied minority d-states were shifted to high energy regions and that cross the Fermi level, thus destroy the surface half-metallicity. On the other hand the surface states at the surface of the TiSn-term were located just below the Fermi level, which reduces the minority spin-gap with respect to that of the center layer. The calculated magnetic moment of the surface Co atom for the Co-term was increased by 10 % to 1.16 ${mu}_B$ with respect to that of the inner-layers, while the magnetic moment of the subsurface Co atom in the TiSn-term has almost same value of the innerlayers(1.03 ${mu}_B$).