Past exam of the mathematics course of the University of Cambridge / 2002 / iii / Paper 22 / 7 / v / Solution

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.