We give a local description of the topology of the space of all geometric limits of closed abelian subgroups of PSL2(C). More precisely, we give geometric descriptions for all possible neighborhoods of a point of this space. Intuition from hyperbolic geometry plays an important role by identifying PSL2(C) with the group of isometries of H-3. The tools and ideas developed in the authors' previous paper on one-generator closed subgroups of PSL2(R) allow one to reduce this problem to a problem about the geometric limits of certain closed subgroups of C and C*.