I'm trying to study through Alain M. Robert's A Course in p-adic Analysis and these 2 questions from the excercises of chapter 1 have me stumped. They're questions 12 and 13 on page no. 65 of the book. Can anybody assist me?
Let $E$ be a compact metric space, $\mathcal{M}_2$ the free monoid generated by two letters, say $0$ and $1$, and $\mathcal{P}(E)$ denotes the set of parts (power set) of $E$. I have to show that for any map $\phi:\mathcal{M}_2 \rightarrow \mathcal{P}(E)$ having the properties:
(a)$\phi(\emptyset)=E,\ \phi(w)=\phi(w0)\cup\phi(w1)\ \ (w \in \mathcal{M}_2)$
(b) $\delta(\phi(w_n))\rightarrow 0$ when the $w_n$ are the initial segments of an infinite word
(c) $\bigcap(\phi(w_n))\neq 0$ when the $w_n$ are the initial segments of an infinite word
there exists a continuous surjective map $$f:\mathbb{Z}_2 \rightarrow E\ \ \text{such that } f(B_w) = \phi(w)$$
And using the above result in some way, I need to show that in general when given a compact metric space $E$, there exists a continuous surjective map $f:\mathbb{Z}_2 \rightarrow E$, therefore deducing that $\mathbb{Z}_p$ is homeomorphic to $\mathbb{Z}_2$, for all $p \neq 2$
In fact, $\Bbb{Z}_p$ is homeomorphic to $\Bbb{Z}_q$ for all primes $p$ and $q$. For the proof, see for example Lemma $8$, $9$ and $10$ here, in the nice article by Scott Zinzer.