Half-metallicity due to the coexistence of metallic nature for one spin component and insulating nature for the other is a base of spintronics devices, but was only achieved in few materials. From first-principles calculations, we demonstrate that a recently-synthesized two-dimensional organometallic framework of 1,3,5-tris(pyridyl)benzene and Cu atoms (Cu-TPyB) has robust half-metallicity. High electron velocity in one spin channel at Dirac point and a relatively large band gap in the other make the material meeting the demand of filtering the current into a single spin component. Moreover, spin-orbit coupling induces topologically nontrivial band gaps in the vicinity of the Fermi level, which are implementable for achieving quantum anomalous Hall effect in a low temperature range (<8 K).