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\label{thm:compactness}
The topological space $\tau^3$, equipped with
the cylinder topology
(Theorem~\textup{\ref{thm:cylinder-basis}}),
is compact.

The inverse-limit construction provides the proof.
By definition (Chapter~\ref{ch:tau-admissible-points}),
\[
    \tau^3
    \;=\;
    \varprojlim_{k}\, \mathbb{Z}/P_k\mathbb{Z},
\]
where $P_k = p_1 \cdot p_2 \cdots p_k$
is the $k$th primorial
and the transition maps
$\pi_{k,l} : \mathbb{Z}/P_l\mathbb{Z}
\to \mathbb{Z}/P_k\mathbb{Z}$
are the canonical reductions
(I.T18, Book~I).

Each factor $\mathbb{Z}/P_k\mathbb{Z}$
is a finite set with $P_k$ elements,
hence compact and discrete.
By Tychonoff's theorem,
the product
$\prod_{k \geq 1} \mathbb{Z}/P_k\mathbb{Z}$
is compact.
The inverse limit $\tau^3$ is a closed subset
of this product:
it is defined by the equational conditions
$\pi_{k,l}(x_l) = x_k$ for all $k < l$,
and each such condition cuts out a closed subset
(preimage of the diagonal
under the continuous projection).
A closed subset of a compact space is compact.

Hence $\tau^3$ is compact.