The Sun's atmosphere contains many diverse phenomena that are dominated by the coronal magnetic field. To understand these phenomena it is helpful to determine first the structure of the magnetic field, i.e., the magnetic topology. We study here the topological structure of the coronal magnetic field arising from the interaction of two bipolar regions, for which we find that four distinct, topologically stable states are possible. A bifurcation diagram is produced, showing how the magnetic configuration can change from one topology to another as the relative orientation and sizes of the bipolar regions are varied. The changes are produced either by a global separator bifurcation, a local double-separator bifurcation, a new, global separatrix quasi-bifurcation, or a new, global spine quasi-bifurcation.