# Unique Way to Extend a Sheaf by 0? Serre's FAC

I am working on understanding the proof of the following proposition given in Serre's FAC. The following is verbatim from page 11.

Proposition 6. Let $$Y$$ be a closed subspace of $$X$$, and let $$\mathscr{G}$$ be a sheaf on $$Y$$. Put $$\mathscr{F}_{x} = \mathscr{G}_{x}$$ for $$x \in Y$$, and $$\mathscr{F}_x = 0$$ for $$x \notin Y$$, and let $$\mathscr{F}$$ be the sum of the sets $$\mathscr{F}_x$$. Then $$\mathscr{F}$$ admits a unique structure of a sheaf over $$X$$ such that $$\mathscr{F}(Y)=\mathscr{G}$$.

Proof (Uniqueness). Let $$U$$ be an open subset of $$X$$; if $$s$$ is a section of $$\mathscr{G}$$ on $$U \cap Y$$, extend $$s$$ by 0 on $$U$$-$$(U \cap Y)$$. When $$s$$ runs over $$\Gamma(U \cap Y, \mathscr{G}$$), we obtain in this way a group $$\mathscr{F}_U$$ of mappings from $$U$$ to $$\mathscr{F}$$. Proposition 5 then shows that if $$\mathscr{F}$$ is equipped with the structure of a sheaf such that $$\mathscr{F}(Y) = \mathscr{G}$$, we have $$\mathscr{F}_{U} = \Gamma(U,\mathscr{F})$$ which proves the uniqueness of the structure in question.

Proposition 5. If a sheaf $$\mathscr{F}$$ is concentrated on a closed subspace $$Y$$, the homomorphism $$\rho_{Y}^{X}\colon \Gamma(X,\mathscr{F}) \to \Gamma(Y,\mathscr{F}(Y))$$ is bijective.

My understanding of the assertion in proposition 6. What we are saying is that for a top space $$X$$, and a closed subspace $$Y$$, if we have some sheaf $$\mathscr{G}$$ on $$Y$$ then there is exactly one way to extend the sheaf by 0 such that restricting back to $$Y$$ recovers the sheaf $$\mathscr{G}$$. Since I am only focusing on the uniqueness, what this should mean is that if I have $$\mathscr{F}_1$$ and $$\mathscr{F}_2$$ that are constructed as the sum of the sets $$\mathscr{F}_x$$ as described, then if I augment $$\mathscr{F}_1$$ and $$\mathscr{F}_2$$ with sheaf structures (a topology that makes the projection map a local homeomorphism, and group operations continuous) such that restricting to $$Y$$ gives $$\mathscr{G}$$, then $$\mathscr{F}_1$$ and $$\mathscr{F}_2$$ are isomorphic, which should mean the existence of a homeomorphism between them that commutes with projection maps?

My understanding of Serre's proof. Basically nothing, in particular how is he applying proposition 5? It seems he is showing that $$\Gamma(U,\mathscr{F})$$ is uniquely determined, but can someone elaborate on his argument for me?

Slightly unrelated question: In the situation that $$\mathscr{F}_{x} = \mathscr{G}_{x}$$ for $$x \in Y$$, and $$\mathscr{F}_x = 0$$ for $$x \notin Y$$, and let $$\mathscr{F}$$ be the sum of the sets $$\mathscr{F}_x$$, can someone give an example of a sheaf structure on $$\mathscr{F}$$ so that $$\mathscr{F}(Y)$$ is not $$\mathscr{G}$$?