I am learning about topological groups, and keep getting confused when I try to be precise about exactly what the structure is. I have also seen that most textbooks omit a specific topology on a group altogether, simply stating that it is a topological group.
For example, in Munkres' 'Topology', there are supplementary exercises asking you to prove that $(\mathbb{R}, +)$ is a topological group, but no topology is specified.
Do we always assume that the topology on a group is the most "obvious" one? Do certain groups have a property where the group operation map and inversion map are always continuous, regardless of the topology?
I hope my question is clear enough. Thanks