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\label{thm:totally-disconnected}
The topological space $\tau^3$ is totally disconnected:
the only connected subsets of~$\tau^3$ are singletons.

Let $S \subseteq \tau^3$ contain two distinct points
$x, y \in S$ with $x \neq y$.
We show that $S$ is disconnected.

Let $\delta(x,y) = k$.
The stage-$(k{+}1)$ cylinder $C_{k+1}(x)$
is a clopen set
(Definition~\ref{def:clopen-basis},
Proposition~\ref{prop:ch09-complement},
Chapter~\ref{ch:cylinder-domains}).
Define
\[
    U \;:=\; S \cap C_{k+1}(x),
    \qquad
    V \;:=\; S \setminus C_{k+1}(x)
    \;=\; S \cap \bigl(\tau^3 \setminus C_{k+1}(x)\bigr).
\]
Since $C_{k+1}(x)$ is clopen in~$\tau^3$,
both $U$ and $V$ are open in the subspace topology on~$S$.
We have $x \in U$ (since $x \in C_{k+1}(x)$)
and $y \in V$ (since $\pi_{k+1}(y) \neq \pi_{k+1}(x)$
implies $y \notin C_{k+1}(x)$).
Furthermore, $U \cap V = \varnothing$ and $U \cup V = S$.
Hence $S$ is disconnected.

Since every subset with more than one point
is disconnected,
the connected components of~$\tau^3$ are singletons.