> [!theorem] Lee 2.25 > > Let $X$ be a [[n-Manifold|n-manifold]], $A \subset X$ be a [[Closed Set|closed]] set, and $U \supset A$ be an [[Open Set|open]] [[Neighbourhood|neighbourhood]], then there exists a smooth [[Function on Manifold|function]] $f: X \to \real$ such that $f|_{A} = 1$ and $f|_{U^c} = 0$. > > *Proof*. Let $U_0 = U$ and $U_1 = X \setminus A$, then there exists a smooth partition of unity $\bracs{\psi_0, \psi_1}$ subordinate to the open cover. Since $\psi_1 = 0$ on $A$, $\psi_0 = 1$ on $A$, and vanishes outside of $U$, which gives the desired bump.