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I know group theory and I'm familiar with the concept and definition of Group. Today I was reading an article about topology and discoverer the concept of "topological group". I read the definition and immediately the following question came to my mind:

What is the purpose of topological groups? What can we do with topological groups that we can't do with regular groups? What are topological groups used for? What can topological groups do that groups without a topology can't?

I barely know any topology and perhaps that's why I can't see any utility of adding a topology to a group.

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  • $\begingroup$ For example Lie groups (which are special topological groups) are used widely in the theory of differential equations. In general topological groups are studied simply because most (if not all) naturally occuring groups come together with a natural topology. $\endgroup$
    – freakish
    Commented Aug 5, 2020 at 13:48
  • $\begingroup$ In the case of finite groups, there is little or no topology involved, but for familiar infinite groups such as $\,\mathbb Q,\mathbb R,\mathbb C$ and so on, they come with natural topologies such that the group operations are continuous and hence, making them topological groups. $\endgroup$
    – Somos
    Commented Aug 5, 2020 at 14:29

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Well... you clearly know some topological groups: Take the fields $\mathbb{Q},$ $\mathbb{R}$ and $\mathbb{C}$ for instance, or the multiplicative group $(0,\infty)$. Now ask yourself what sort of homomorphisms you know involving these groups. The first to come to mind are probably linear maps, the exponential function or the logarithm. Now, these all have the property of being more than continuous - they're even smooth.

Do discontinuous homomorphisms exist? Well, yeah, just check https://en.wikipedia.org/wiki/Cauchy%27s_functional_equation#Existence_of_nonlinear_solutions_over_the_real_numbers. However, you've never heard of those, because it's not clear that they're interesting. It's something of a pathology of the real numbers that its endomorphisms aren't all $\mathbb{R}$-linear (although, they are all $\mathbb{Q}$-linear).

Why are they not interesting? Well, the same reason why you probably cannot name many maps $\mathbb{Z}\to \mathbb{Z}$ that aren't either group homomorphisms or have some combinatorial interpretation. Generally, we care about the mathematical structure of our objects. Thus, when looking at maps of groups, we are mostly interested in the maps that can see the group structure - these are the homomorphisms.

Similarly, if you are looking at a topological space, the most natural maps to care about are those which respect the topological structure - i.e., the continuous maps. So I would say that the point is more so that a bunch of groups that we naturally study also happen to carry natural topologies - see the above number fields and all manners of matrix groups. Since they are both groups and topological spaces in a natural fashion, it makes sense to study those two properties at the same time.

Now, of course, there is strength to be had in topology. For instance, you get the intermediate value theorem for the real numbers and more generally, that continuous homomorphisms have to respect connectivity properties. The regularity of continuous maps is very rich, I just don't think that that in and of itself is the reason why topological groups are interesting. They are just interesting because lots of interesting groups happen to be topological.

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