In the screenshots attached above George Bergman outlines his way of proving $HSP \ne SHPS$
I understand the first definition as the group of affine transformations and each element of the group can look like $(a,b)$ where a corresponds to rotation and b for translation.
In the second definition I have two issues:
- Is the $n$ fixed or can it vary for each different $G_p$ ?
- I can not see how to prove that $G_p$ is dense in G. Some hint in this direction will solve my problem. (I have minimal understanding of topology so hint can be more direct. )
About the first question:
$n$ varies for each $G_p$, and that's the reason for which $G_p$ is dense in $G$.
For example, taking $p=2$, you get rotations $0$ and $\pi$ (it's not clear in the part of the screenshot, but I suppose $0 \leq m < p^n$), for $n=1$; then $0, \pi/4, \pi$ and $3\pi/4$, if $n=2$, and so on...
The same for other primes.
About the second question:
$G_p$ is dense in $G$ in the same sense that, for example, $\mathbb{Q}$ is dense in $\mathbb{R}$.
More precisely, we're approximating members of the interval $[0, 2\pi]$ by sequences of members of sets of the form $$\left\{ \frac{2\pi m}{p^n} : n \in \mathbb{N}, 0 \leq m < p^n \right\},$$ for each prime number $p$.
Again, considering $p=2$, and fixing $\alpha \in [0,2\pi]$ we can build such sequence as follows.
For $n=1$, we choose the value of $a_1$.
Let $a_1 = 0$ if $\alpha \leq \pi$; otherwise $a_1 = 1$ (this is covered in the displayed set above by taking either $m=0$ or $m=1$).
For $n=2$, we choose $a_2$.
If $a_1=0$, we then take $a_2=0$, if $\alpha \leq \pi/2$; otherwise, $a_2=1$ (here we're considering $m=0$ or $m=1$).
If $a_1=1$, we then take $a_2=0$, if $\alpha \leq 3\pi/2$; otherwise, $a_2=1$ (here we're considering $m=2$ or $m=3$).
Go on like that until you have a sequence $(a_n)_{n \in \mathbb{N}}$, and then, $$\alpha_n = \sum_{k = 1}^n 2a_k\pi$$ is such that $\lim_{n \to \infty} \alpha_n = \alpha$.