Let $f:X\rightarrow Y$ be a continuous map of topological spaces, and $\mathscr{G}$ a sheaf on $Y$. So far I failed to come up with a simple example where the presheaf $f^{-1}\mathscr{G}$ on $X$ obtained via the direct limit $$f^{-1}\mathscr{G}(U):=\lim_{f(U)\subset V}\mathscr{G}(V)$$ is not a sheaf. If anyone could give me an example (the simpler the better), it would be greatly appreciated.
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Take $X = $ two points with the discrete topology and $Y = $ a point. Let $\mathscr{G}$ be a set with more than two elements, sitting on $Y$. What you get is the corresponding constant pre-sheaf on $X$, which fails to satisfy the gluing axiom.
TTS
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Can you think of an example where $X$ is a subset of $Y$ ? If it is an open subset, then $\mathscr{G}|_X$ is also a sheaf. – ashpool Jul 09 '13 at 16:12
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Hm, I'll try to think of one. – TTS Jul 09 '13 at 16:21