I would like to consider three sheaves $\mathcal{C}^0$, $\mathcal{H}$ and $\mathcal{S}$ on $\mathbb{C}$ (endowed with the euclidean topology): the first is the sheaf of continuous $\mathbb{C}$-valued functions, the second is the sheaf of holomorphic functions and the third is the sheaf of "overconvergent analytic functions" of radius $1$, by which I mean the subsheaf of $\mathcal{H}^0$ assigning to each **edit:** *connected $U$ the set*

$$ \{f\in\mathcal{H}^0(U)\bigm\vert\forall\;x\in U,\text{ the Taylor series of }f\text{ converges in }B(x,1)\}. $$

By analytic continuation, given an open $U\subseteq \mathbb{C}$, if two sections $s_1,s_2\in \mathcal{S}(U)$ coincide everywhere but possibly in a small ball of radius $\rho<1/2$, they must in fact coincide on all $U$ and hence be equal; but this is false if $s_1,s_2\in\mathcal{C}^0(U)$. To me, it looks as if $\mathcal{S}$ is a sheaf for a finer Grothendieck topology (on the category $\operatorname{Top}(\mathbb{C})$), where we declare a collection of opens to cover $U$ even if their union misses a small ball inside $U$. I must say that my motivation is more to see an example of a Grothendieck topology on the category of opens of a "street-man topological space" (forgetting Zariski, étale, fpqc and the like) rather than understanding something new on analytic functions.

I have been trying hard to define a Grothendieck topology different from the "usual" one given by all topological covers on the category $\operatorname{Top}(\mathbb{C})$ and got nowhere: a bestiary of my missed trials is (I refer to this Wikipedia page for axioms T1) and T2))

- Covering sieves of an open $U$ are attached to collections of opens which, when adding to the collection a family of disjoint balls $B(x,1)$ (disjointness is crucial to still have a sensible sheaf theory), become a true cover of $U$: it verifies axiom T1) but not T2);
- Same as above but with balls $B(x,\delta)$ with $\delta$ smaller than the diameter of $U$: it verifies T2) but not T1);
- Covering sieves of $U$ are attached to collections of open balls $B(x,1/2)$ such that the collection $B(x,1)$ covers $U$: it verifies T1) but not T2);
- Same as above replacing $1/2$ by $\delta=$diameter of $Y$: verifies T2) but not T1).

I ended up in despair. I'd like either to know if the unique Grothendieck topology on $\operatorname{Top}(\mathbb{C})$ is the "usual one"; or to finally find an example of one a such. Secondly, if possible, I would also like to understand if my idea that analytic continuation can be rephrased in saying that $\mathcal{S}$ is a sheaf for a finer topology than the usual one, makes sense and if it is true or false.

*Addendum* After a conversation with MO user ACL regarding this question I convinced myself that one can define a topology by declaring that only sieves attached to countable covers (or, I believe, finite covers) are covering sieves. I am unable though to produce an interesting example of a sheaf for either of these topologies which is not a sheaf for the usual one. So I'd be happy with either

- new examples of Grothendieck topologies, or
- with an example of a "natural" sheaf for one of the two above, or
- with something about my original question about $\mathcal{S}$.