Is there a name for the property of a sheaf $\mathcal F$ such that the restriction maps $\mathcal F(V) \to \mathcal F(U)$ are injective when $V$ is connected and $U$ is nonempty?


In other words, this is a sheaf which satisfies the [identity theorem](http://en.wikipedia.org/wiki/Identity_theorem) of complex analysis (taking for $\mathcal F$ the sheaf of holomorphic functions). 


Many sheaves in geometry satisfy this property, but I've never heard a name for it. Perhaps a "rigid" sheaf, in contrast to a "flasque" sheaf? (This rigidity property seems to be the one which gave [rigid analytic geometry](http://en.wikipedia.org/wiki/Rigid_analytic_space) its name, but I am not completely sure about that, and in any case, I have never seen the words "rigid sheaf" written down.)


Also, if these sheaves do not have a name, I would be curious to know why that might be. Perhaps the property is not useful enough on its own to deserve a name?


Thank you.



Is there a name for a "rigid" sheaf?
The problem is that your definition is well behaved only if there is enough open subsets $V$ such that $V$ is connected (if there is no such open subset, then your condition is empty) hence the notion as you defined is well behaved only on a locally connected space. 


Once you have fixed this issue (either by slightly changing your definition, or by restricting yourself to locally connected space) The name your are looking for is *"decidable sheaf"*.


In general, an object $X$ of a topos is called decidable (or is said to have a decidable equality) if the diagonal embedding $X \rightarrow X \times X$ is complemented, that is if there exists a sub-object $\Delta^{c}$ of $X \times X$ such that $X \times X \simeq X \coprod \Delta^c$ (with the $X$ part corresponding to the diagonal inclusion).


In the case of the topos of sheaves over a topological space $X$ the sheaf $\mathcal{F}$ is decidable iff for each couple $f,g \in \mathcal{F}(U)$ the (open) set $W$ of points $x \in U$ such that $f$ and $g$ are equal in the stalks at $x$ is closed in $U$.


You can easily check that this imply the property you want and that on a locally connected space they are equivalent.


Also note that this is equivalent to the fact that the map from the etale space $Y$ of $\mathcal{F}$ to $X$ is relatively hausdorff in the sense that the diagonal embeddings $Y \rightarrow Y \times\_X Y$ is a closed embeddings.


Also this version of "rigidity" is a local property, and stable by arbitrary pullback, whereas I don't think the original definition is.


