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Let $X$ be a topological space containing two closed points $x,y$ and let $i : \{x,y\} \to X$ denote the inclusion map. Notice that $\{x,y\}$ carries the discrete topology. Let $F$ be a sheaf on $X$. Then $i^{-1} F$ is a presheaf on $\{x,y\}$ which is given by $(i^{-1} F)(\emptyset)=1$ (the terminal set), $(i^{-1} F)(\{x\}) = F_x$ (the stalk at $x$), $(i^{-1} F)(\{y\})=F_y$ (the stalk at $y$) and $$(i^{-1} F)(\{x,y\}) = \varinjlim_{ U \subseteq X \text{ open}, ~ x,y \in U } F(U).$$ Why is, $i^{-1} F$ a sheaf if and only if the canonical map $$\varinjlim_{ U \subseteq X \text{ open}, ~ x,y \in U } F(U) \to F_x \times F_y.$$ is an isomorphism ?.

Thanks a lot to all of you.

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