We present a study based on Density Functional Theory (DFT) of the volume and two surfaces (0001) of MgB2, one of them terminated in Mg and the other one terminated in B. Each one of the surface was relaxed and their electronic properties were determined. From calculation of the enthalpy of formation we found that the Mg-terminated surface is energetically favored. The bands seem to present a formation similar to the Dirac's cone as that are presented in graphene, but in MgB2 is above of the Fermi level. In the three cases, volume and the two surfaces, the behaviour is boron-metallic, because there are strong presence of B orbital's in the neighborhood of the Ferm level.