In Serre's FAC, on page 9 (http://achinger.impan.pl/fac/fac.pdf), he states that "One verifies immediately that the data (a) and (b) satisfy the axioms (I) and (II)."
It's clear to me why the map $f\to-f$ is continuous, but I don't see why $(f,g)\to f+g$ (from $\mathcal{F}+\mathcal{F}\to\mathcal{F}$) is.
I also don't really understand the motivation behind the open sets of $\mathcal{F}$ -- why do we want to take $[t,U]$ (as defined in axiom (b) of page 9) to be the generators?
Edit: I thought about this a bit more and was wondering if the following was correct: If we let $\phi$ be the map that takes $(f,g)\mapsto f+g$, is it true that $\phi^{-1}([t,U])=\bigcup_{t_1+t_2=t}\left([t_1,U]\times[t_2,U]\cap(\mathcal{F}+\mathcal{F})\right)$? If so, this is a union of open sets, which is an open set, and proves the continuity of $\phi$ (we let $[t,U]$ be any neighborhood basis element) ?
Let's think about why we might want to define things that way, with the basis of open sets $[t,U]$. First off, it might help be thinking of elements of $\mathscr F_U$ as functions on the open set $U \subset X$ and the $\phi^U_V$ maps as restriction of functions. Then $\phi_x^U(t)$ is the "germ" of the function $t$ at $x$. A germ at $x$ is an equivalence class of functions defined near $x$ with two functions being equivalent if they agree provided you zoom in enough. You might google germs of functions if that's new to you. Now, back to the topology defined on $\mathscr F$. It lets us describe the open sets of $\mathscr F$ in the following (nice) ways. I'll just say it a few different ways and hope one sticks.
(1) A set $W \subset \mathscr F$ is open if and only if for each $g\in W$, there is an open set $\pi(g) \in U\subset \pi(W) \subset X$ and a function $f \in \mathscr F_U$ so that $\phi_{\pi(g)}^U (f) = g$ and $\phi_x^U(f) \in W$ for all $x\in U$. (Okay, that's not so nice yet -- let me try again.)
(2) $W$ is open if and only it is locally (over $X$) the set of germs of a single function.
(3) $W$ is open if and only if there exist open sets $U_i \subset X$ (ranging over $i \in I$, some indexing set) and functions $f_i^{U_i} \in \mathscr F_{U_i}$ so that $W$ is the union of the sets of germs of the $f_i^{U_i}$, i.e., the union of the sets $\{\phi_x^{U_i}(f) : x\in U_i\}$.
For me, thinking of things this way makes it clear that (I) and (II) hold. But maybe you'll figure it out a different way.