A tale of three varieties

A long time ago, in a galaxy far, far away, there were three distinguished and well-studied classes of topological groups. However, little was known about the three varieties of topological groups they generate. This thesis tells the tale of these varieties. It becomes apparent that the three varieties are all distinct and that one is in fact the variety of abelian topological groups, that is, their entire universe. The other two are comparatively small, however, no less important and interesting. A framework is established within which the three varieties are compared with each other and also come up against the varieties generated by other important classes of topological groups. Discoveries are made that will stun even the most brave....