Non-Archimedean Division Ring is Totally Disconnected

Theorem
Let $\struct{R, \norm{\,\cdot\,} }$ be a non-Archimedean normed division ring.

Let $\tau$ be the topology induced by the norm $\norm{\,\cdot\,}$.

Then the topological space $\struct{R, \tau}$ is totally disconnected.

Proof
Let $S$ be a subset of $R$.

Let $x, y \in S: x \ne y$

Let $r \in \R_{>0} : r = \norm {x - y}$

Consider the open ball, $B_r \paren{x}$, then:
 * $x \in B_r \paren{x}$
 * $y \notin B_r \paren{x}$

By Open Balls are Clopen then $B_r \paren{x}$ is both open and closed.

By Complement of Clopen Space is Clopen then $R \setminus B_r \paren{x}$ is open.

Hence $S$ is not connected.

The result follows.