I'm currently studying topological groups, and I'm just wondering why we need continuity of the binary operation and the inverse map in the definition. What falls apart if we omit this condition? Why can we not just equip the group with any topology?
I know that if the inverse map isn't continuous then it is a paratopological group.
If possible, could you point me in the direction of resources that tackle this question?
Think of it this way. If you define a topological group to be merely a group with a topology on it, the master text of all results in the theory would simply be the master text of group theory results bound on top of the master text of all topological results. There is nothing new, no new knowledge is created. However, if you demand that the group structure interact with the topology, then suddenly you can prove new things about these objects (homogeneity of the topology, for one). Now the master text of all results is much more than the union of the texts for group theory and topology. The point is that the additional requirements give you more stringent hypotheses, allowing you to prove new results that weren't necessarily true otherwise.
Also, if you want evidence of all this, do a literature search for what you call a "paratopological group" and compare it to what you get for "topological group."