Compatibility of group structure and topological structure for topological groups

Question

I am fairly new to the concept of topological groups, and would like to understand the underlying idea. My question is about the compatibility between the two structures. The definition of a topological group requires that the topology on $G$ makes the multiplication and inversion maps continuous. However, I am wondering if there is an example of a topological group that does not have some topology a priori.

All the examples I know (i.e. $\mathbb{C}$, $\mathbb{R^*}$, $\textrm{GL}_n(\mathbb{R})$, etc.) already have a topological structure, and I would like to know how the group structure affects what topologies are possible on $G$. For example, if we are given an arbitrary group $G$, and we would like to show it is a topological group, the only way I know how for now is via the definition. This seems like circular reasoning to me because it requires that we know what open subsets of $G$ are, and in order to do so we must define a topology on $G$. But to define a topology on $G$ we need to check that it makes the multiplication and inversion maps continuous, and so on...

More importantly, I have studied these objects in very separate settings (group/ring/field theory, and now topology) and I was wondering if there is a good way to unite common ideas. For now, I am having a tough time using my intuition from algebra to apply to questions regarding the topological structure of groups. I tried defining a topology on $D_6$ but got lost because of the above.

Answer 1

Here is an exercise: suppose $G$ is a finite group. Show that picking a topology on it making it a topological group is equivalent to picking a normal subgroup $N$ of it, which is the closure of the identity in the topology (and this completely determines the topology). In particular, the only Hausdorff topology on $G$ making it a topological group is the discrete topology.

This seems like circular reasoning to me because it requires that we know what open subsets of $G$ are, and in order to do so we must define a topology on $G$. But to define a topology on $G$ we need to check that it makes the multiplication and inversion maps continuous, and so on...

There's nothing circular here. First you define a topology and then you define a group structure. Then you can check whether they're compatible. Sometimes the answer is yes and sometimes the answer is no.