What is a topological group intuitively, beyond just being able to say things are near to each other in a group, and why is it a good idea to consider this theory as part of general topology as Bourbaki did?
For example, what does it really mean for addition and multiplication on $\mathbb{Z}$ and $\mathbb{R}$ to be continuous? Is there a geometric motivation behind these examples, similar to the way there is for Lie groups? The proofs for these examples are easy, but it just seems like a useless exercise in formalism and definitions simply for the sake of formalism, with no actual meaning...
I know the theory originated from Lie's theory of differential equations, and can bluff motivation by thinking of the continuous symmetries in Noether's theorem, but that vague image has no direct relationship to the axioms of a topological group:
which I hope you could motivate and give a bit of life to (i.e. exactly how this allows us to talk about near-ness in a group and why this way), as part of your answer.
Thanks!
To make things brief, I'll first consider metrizable groups (groups in which the topology is induced by a metric), e.g. $\Bbb R^n$ with Euclidean metric or $\Bbb Z^n$ with discrete metric. We have (at least) two structures we could put on these sets: the topological and group structures. Treating these independently is all well and good but it might be nice to mix them together in some way. A metric topology is concerned with convergence of sequences and groups are concerned with their group operations of multiplication and inversion. This suggests that somehow we need our group operations to play with the topology on our group.
Suppose we are working in $\Bbb R$ and have two sequences of real numbers $(x_n)_{n\in\Bbb N}$ and $(y_n)_{n\in\Bbb N}$ converging to $x$ and $y$, respectively. We would like to be able to say that $(x_n+y_n)_{n\in\Bbb N}$ converges to $x+y$ since this would agree with our intuition about convergent sequences. This is easy to see for $\Bbb R$ because we have the triangle inequality at our disposal. We have that
$$\|(x_n+y_n)-(x+y)\| \le \|x_n-x\| + \|y_n-y\|.$$
But since $x_n\to x$ and $y_n\to y$, the left hand side is as small as we want so addition is continuous, meaning that if $x_n\to x$ and $y_n\to y$, then $x_n+y_n\to x+y$. Similarly, if $x_n\to x$, then we know that $-x_n\to-x$ so we would say that the operation of inversion is continuous.
$\Bbb R$ serves as the prototype for topological groups in general since it is perhaps the nicest example that isn't trivial (i.e. finite or the topology is discrete). So with this in mind, in order to interlace our topology and group operations we might ask that multiplication and inversion be continuous operations. Continuity means can mean a few different things (depending on the language in which you want to write it).
If you are aware, we could define continuity in terms of nets which behave a lot like sequences in some ways (but very differently in others.. so beware) and you can think of them as a generalization of sequences. Sequences are a discrete phenomenon (functions on $\Bbb N$) whereas nets can take any indexing set (could be an uncountable set). In this setting what we would ask is that if $\Lambda$ is an indexing set, $(x_{\lambda})_{\lambda\in\Lambda}$ is a net converging to $x$ and $(y_{\lambda})_{\lambda\in\Lambda}$ is a net converging to $y$, then $x_{\lambda}y_{\lambda}$ converges to $xy$. Similarly we would require that $x_{\lambda}^{-1}$ converges to $x^{-1}$.
Alternatively, to avoid the headache of nets, we can just talk about open sets. The theme of topology is to avoid the "discrete" (sequence-based) nature of metric spaces. In metric spaces, we frame convergence in the language of open balls. In topological spaces, we frame convergence in terms of just open sets.
Suppose we have a collection of elements $x_{\lambda}$ in our group. We would say that $x_{\lambda}$ converges to an element $x$ if for any open neighborhood $N_x$ of $x$, the tail end of $x_{\lambda}$ lives inside of $N_x$. This observation allows us to shed the idea of limits and just talk about open sets. The reason being that if $y_{\lambda}$ converges to $y$, then we could find an open neighborhood $N_y$ of $y$ so that the tail end of $y_{\lambda}$ lives inside of $N_y$. If $x_{\lambda}\in N_x$ and $y_{\lambda}\in N_y$, then $x_{\lambda}y_{\lambda} \in N_xN_y$. We would say that $x_{\lambda}y_{\lambda}\to xy$ if we can find an open neighborhood $N$ of $xy$ so that the tail end of $x_{\lambda}y_{\lambda}$ lives inside of it. Meaning.. if we can find $N_x,N_y$ so that $N_xN_y\subseteq N$. (This is secretly the same as the nets-based definition since the way we define convergence of nets is by.. you guessed it.. open sets.)
Now then, this is all well and good but what about an example? I'll do a perhaps weird example that isn't a topological group. We can't really mess around with the group's operations but we can mess with the topology we decide to put on the group. Let's consider $\Bbb R^2$ and let's consider the topology generated by the following metric:
$$d(x,y) = \begin{cases} 1 & x,y\,\,\text{linearly independent} \\ \|x-y\|_2 & x,y\,\,\text{linearly dependent}\end{cases}.$$
You can think of this as decomposing $\Bbb R^2$ into uncountable copies of the real line where each copy is a ray emanating from the origin such that each copy is distance $1$ apart. Suppose $x_n\to 0$, then it's not hard to see that the $x_n$ need only to converge to $0$ in the usual sense. Alternatively, if $x_n$ converges to something not zero, then eventually the $x_n$ must lie on the same line. (This is where the discrete topology comes into play.)
(In this paragraph we assume $x,y\neq 0$.) Suppose $x_n\to x$, then for some $N_1$ and all $n>N_1$, all of the $x_n$ lie on the same line. Similarly suppose $y_n\to y$, then for some $N_2$ and all $n>N_2$, all of the $y_n$ lie on the same line. Take $n = \max\{N_1,N_2\}$. Then we have that $d(x_n+y_n,x+y) = \|x_n+y_n-(x+y)\|$ since they lie on the same line and this goes to zero from earlier notions. Likewise it's not hard to see that $x_n^{-1} = -x_n$ converges to $-x$. Everything seems in order here.
What if one converges to zero but the other does not? Suppose $x_n = x \neq 0$ so $x_n$ trivially converges to $x$ and suppose that $y_n\to 0$ but $y_n$ spirals around $0$ (take a discretized Archimedes spiral). Then $x_n+y_n$ will not lie on the same line as $x$ and so $x_n+y_n$ will not converge to $x$ as we would like. Thus this does not make for a topological group since addition is not a continuous operation.