Is assuming the gluing axiom for any pair of sections equivalent with gluing arbitrary collections of sections?

Question

In Griffiths and Harris the gluing axiom for sheaves is given as

For any pair of open set $U,V$ and sections $\sigma\in\mathcal{F}(U)$, $\tau\in\mathcal{F}(V)$ s.t. $\tau|_{U\cap V}=\sigma|_{U\cap V}$ there exists a section $\rho\in \mathcal{F}(U\cup V)$ with $\rho|_U=\sigma$, $\rho|_V=\tau$

On wikipedia (and other places) the axiom is given as

If $\{U_i\}$ is an open covering of an open set $U$, and if for each $i$ a section $s_i \in \mathcal{F}(U_i)$ is given such that for each pair $U_i,U_j$ of the covering sets the restrictions of $s_i$ and $s_j$ agree on the overlaps: $s_i|_{U_i\cap U_j} = s_j|_{U_i\cap U_j}$, then there is a section $s\in \mathcal{F}(U)$ such that $s|_{U_i} = s_i$ for each $i$.

To me the definition on wikipedia seems a lot stronger. The Griffiths and Harris axiom allows gluing of a finite number of section that agree of overlaps, but I don't see how to generalise it to an arbitrary amount of sections.

Any insight in whether or not these formulations are equivalent and if not why Griffiths and Harris would chose their formulation over the seemingly more common wikipedia formulation.

Answer 1

No, these formulations are not equivalent in general. As an example, consider $\mathbb N$ with the discrete topology, and consider the 'sheaf' of functions with values in $\mathbb C$ that are zero almost always. More precisely, for every $U \subseteq \mathbb N$ we define $$\mathcal F(U) = \{f: U \to \mathbb C: f(n) = 0\textrm{ for all but finitely many $n \in U$}\},$$ with the natural restriction maps. I'll leave it for you to verify (although it is not hard to see) that this gives a sheaf in the sense of Griffiths and Harris but not in the sense of most others. Incidentally, the reason why Wikipedia and most authors do not call the above a sheaf is that the defining property (being zero almost everywhere) is not a local property.

An important case in which the two formulations are equivalent is when the topological space is Noetherian. In that case every open subset is compact so we can always reduce to a finite cover. Noetherian spaces appear a lot in algebraic geometry, so perhaps Griffiths and Harris mean to only study sheafs over those spaces (I don't have the book, so I can't check).

Edit: This MathOverflow post suggest that the definition in Griffiths and Harris is just an error.