%
\label{thm:topology-uniqueness}
Let $\mathcal{T}$ be a topology on~$\tau^3$ satisfying:
\begin{enumerate}
    \item[\textup{(a)}]
          \textbf{CRT continuity.}
          All reduction maps
          $\pi_k : (\tau^3, \mathcal{T}) \to \mathbb{Z}/P_k$
          are continuous
          \textup{(}discrete topology on targets\textup{)}.
    \item[\textup{(b)}]
          \textbf{Hausdorff separation.}
          $(\tau^3, \mathcal{T})$ is Hausdorff.
    \item[\textup{(c)}]
          \textbf{Compactness.}
          $(\tau^3, \mathcal{T})$ is compact.
\end{enumerate}
Then $\mathcal{T}$ equals the profinite topology:
\[
    \boxed{\mathcal{T}
    \;=\;
    \mathcal{T}_{\mathrm{pro}}
    \;:=\;
    \mathrm{Initial}\bigl(\pi_k : k \geq 1\bigr).}
\]

Let $\mathcal{T}_{\mathrm{pro}}$
denote the profinite (cylinder) topology.

\medskip
\noindent\textbf{Step 1.}
By hypothesis~(a), each~$\pi_k$ is $\mathcal{T}$-continuous.
Since $\mathcal{T}_{\mathrm{pro}}$
is the \emph{coarsest} topology
making all~$\pi_k$ continuous
(Proposition~\ref{prop:ch14-initial-topology}),
$\mathcal{T}_{\mathrm{pro}} \subseteq \mathcal{T}$.

\medskip
\noindent\textbf{Step 2.}
The identity map
$\mathrm{id} : (\tau^3, \mathcal{T})
\to (\tau^3, \mathcal{T}_{\mathrm{pro}})$
is continuous by Step~1.
By hypothesis~(c), the domain is compact.
By II.T08, the codomain is Hausdorff.
A continuous bijection from a compact space
to a Hausdorff space is a homeomorphism,
so $\mathcal{T} = \mathcal{T}_{\mathrm{pro}}$.