Experimentally, it is observed that Toroidal Alfvén Eigenmodes (TAEs) are difficult to excite with an external antenna when the plasma is in X-point geometry. Here, the effect of the X-point geometry on the efficiency of the TAE excitation with the external antenna is investigated. In the first part of this paper, the influence of the near-last closed flux surface layer from the core side on the damping of the TAE modes is calculated using the linear resistive eigenvalue MHD code CASTOR. The resistive damping is identified as the main cause of the TAE damping in the open gap in the Alfvén continuum. It is shown that one aspect of the difficulty of excitation of the TAE modes in X-point geometry is an increased damping from the region inside the separatrix. However, the increased damping with the plasma boundary approaching the separatrix is not general and depends on the density profile shape. The second part of this paper discusses the influence of the TAE behavior in the limiter and X-point geometries including the scrape-off layer (SOL) in the reduced MHD code JOREK. It is shown that the dominant effect on the damping of the original TAE mode observed in the limiter configuration is caused by the additional damping in the region of open field lines, i.e., the SOL.