证明不连通性

hbghlyj

Show that $\Bbb Z$ with the 2-adic metric is not connected.

Czhang271828

Step 1. $\{d_p(x,y)\mid x,y\in \mathbb Q_p\}$ is discrete in $(\varepsilon,\infty)$ for each $\varepsilon >0$. Hence each open ball in $\mathbb Q_p$ is also closed.

Step 2. $\mathbb Z_p$ is open/closed ball in $\mathbb Q_p$. Therefore $(\mathbb Z_p,\|\cdot \|_p)$ is a topological subspace of $(\mathbb Q_p,\|\cdot \|_p)$.

Step 3. $\mathbb Q_p$ is totally disconnected. For each distinct $a,b\in \mathbb Q_p$, there exists an open/closed ball $B$ s. t. $a\in B$ and $b\notin B$. As a result, $\mathbb Q_p=B\dot\cup B^c$ is a disjoint union of two open sets.

We deduce that $\mathbb Q_p$ is totally disconnected, so is $\mathbb Z_p$.

hbghlyj

Czhang271828 发表于 2022-7-29 06:49
Step 1. $\{d_p(x,y)\mid x,y\in \mathbb Q_p\}$ is discrete in $(\varepsilon,\infty)$ for each $\varep ...

Thanks for your reply. Please have a look at my work (for $p=2$):

\begin{array}l
B(0,1)=\{y:d_2(0,y)<1\}=\{y:2|y\}\\
B(1,1)=\{y:d_2(1,y)<1\}=\{y:2|y-1\}
\end{array}$B(0,1),B(1,1)$ are disjoint open sets and $\Bbb Z=B(0,1)∪B(1,1)$. So $\Bbb Z$ is not connected with the 2-adic metric.

Is the proof correct?

Czhang271828

hbghlyj 发表于 2022-8-6 10:44
Thanks for your reply. Please have a look at my work (for $p=2$):

\begin{array}l

Exactly correct. Your solution can be easily generalised to $\mathbb Z_p$ topology.

From my point of view, there is no need to explore more on $p$ -adic numbers untill you show some interests in field theory, especially the theory of Krull valuation.

It is highly recommended to learn sth about Monsky's thm, a perfect combination of Sperner's lemma and $p$ -adic theory, which is mainly about the impossibility of dissecting the square into an odd number
of triangles of equal area. You can enjoy

• An exposition of Monsky's theorem, an exploration on Thm 4.5-4.6 is suggested.
• A slide for middle school students.
• The analysis of dissecting the square into an odd number
  of triangles of ALMOST equal area
• Monsky's thm is also mentioned in page 140, Proof's from THE BOOK (4th edition).

hbghlyj

Czhang271828 发表于 2022-8-6 07:12
Monsky's thm is also mentioned in page 140, Proof's from THE BOOK (4th edition).

现在此链接已过期，但网络上有一个存档，网址为 web.archive.org/web/20170316210616/https://pr … fs from THE BOOK.pdf