For the definition of a topological group, I was wondering why the mappings in the definition need to be continuous. Why is this so? Is this just a matter of preference, like it makes them easier to work with? Or is there a more important underlying reason?
I tried looking online and in books to see why the continuity condition is needed, but it never seems to be mentioned. Thanks for the help!
In any nonempty set you can put a group structure: https://math.stackexchange.com/a/105440/17092
If the sets are not finite, you can do it in an infinite number of ways. Just take a bijection from a set to itself, you can transport its group structure to something completely different: the identity can be remapped to any element and any element can be remapped to any other element. So, in a sense, an arbitrary group structure is quite useless.
In any set you can put a topology. All the same reasoning applies here.
Take some complicated bijection $f$ from $\mathbb{Z}$ to itself. Now, you have a strange group structure induced by the usual $+$ and this bijection: $f(a) + f(b) = f(a+b)$. Why is this construction useless if $f$ has no meaningful structure? Notice that $\mathbb{Z}$ is, for example, ordered: $\dotsb < -2 < -1 < 0 < 1 < 2 < \dotsb$.
Those structures are useful because they are somehow related. For example, \begin{equation*} a < b \Rightarrow a + c < b + c. \end{equation*}
The same goes for groups and topology. Not only you have two structures... but they are related to each other.
