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Here is what I am trying to do: Let $X$ be a paracompact smooth manifold. Let $C$ be closed, $U$ open and $C\subset U \subset X$ and $f$ is a smooth map on $U$. I want to show that then there exists an open $V$ and smooth $F$ on $X$ with $F\mid_V = f\mid_V$ and $C\subset V \subset U$. To thi...