Why do Topological Groups Need Continuous Maps? 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!
 A: 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.
