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We prove Booleanity directly from the dense-cover description. Let $B$ be a subsheaf of a sheaf $F$. Its pseudocomplement $N$ consists of sections $x\in F(c)$ such that no restriction along any arrow into $c$ belongs to $B$. This is a subpresheaf. It is also a sheaf: if a section of $F$ restricts into $N$ on a dense covering sieve on a category, any hypothetical restriction into $B$ has a further restriction in that cover; it would then belong to both $B$ and $N$, impossible. Thus the unique amalgamation in $F$ of an $N$-matching family lies in $N$.

For any $x\in F(c)$, the sieve on a category of arrows on which its restriction lies in $B$ or $N$ is dense. Given an arrow $u:d\to c$, either some further restriction belongs to $B$, supplying the required member of that sieve on a category, or none does, in which case the restriction along $u$ itself belongs to $N$. Therefore $B\cup N$ is locally all of $F$, so their join in the sheaf topos is $F$. Their intersection is empty. Every subsheaf consequently has a complement, proving the topos is Boolean.

For two-valuedness, let $U\hookrightarrow 1$ be a nonempty subterminal sheaf. At some stage $m$, its value is the singleton. For every $k\ge m$ there is an arrow $k\to m$, so restriction makes $U(k)$ the singleton as well. At any stage $n$, the sieve on a category of arrows into $n$ whose domains are at least $m$ is dense: after an arrow with domain $l$, choose a further domain $k\ge \max (l,m)$ and use the existence of arrows downwards. On this covering sieve on a category the unique terminal section has a matching family in $U$, so the sheaf condition gives $U(n)=1$. Hence $U=1$. Since the empty sieve on a category never covers, the empty presheaf is an initial sheaf distinct from $1$. Thusand the topos is two-valued, in the distinct global sense of a two-valued topos.

The sheaf $A_n$ is noninitial: it contains $y_n$, whose value at $n$ includes $1_n$. In a two-valued topos its support, the image of $A_n\to 1$, can therefore only be $1$. An image equal to $0$ would force $A_n=0$, since the initial object of a topos is strict. Consequently every $A_n\to 1$ is an epimorphism.

Products of sheaves on a small site exist and are computed as products of the underlying presheaves, since a product of unique matching-family amalgamations again uniquely amalgamates. For the countable product $P={\prod }_{n\in \mathbb{N}}{A}_n$, its value at stage $m$ is ${\prod }_n{A}_n(m)$. The factor with $n=m+1$ is empty by the stronger statement proved in the preceding part. Thus $P(m)=\varnothing$ at every stage $m$, with the unique empty restriction maps. It is the empty presheaf, already an initial sheaf because no empty sieve on a category covers. HenceThis proves the final PDF-only request and exhibits the separating tower site with vanishing product. It is a genuine failure of countable choice in this Boolean two-valued sheaf topos; it does not contradict the classical external construction of the site.

Consider a site indexed by natural-number objects with arrows only downwards, the cancellation condition $fh=gk\Rightarrow f=g$ for equally typed $f,g$, and the separation condition that arrows $f:m\to n$, $g:m\to n+1$ admit $h,k:m+1\to m$ with $fh=fk$ but $gh\ne gk$. Its dense-sieve sheaf topos is a Boolean topos and a two-valued topos. The representables are separated. No nonempty partial map $y_n\to y_{n+1}$ exists, by the separation condition and naturality. The associated sheaves $A_n$ consequently admit no maps $y_n\to A_{n+1}$. Each has nonzero support and maps epimorphically to one, but their product has an empty value at every stage $m$, because its $A_{m+1}$ factor does.