> [!definition] > > Let $X$ be a [[n-Manifold|n-manifold]] of class $C^{p - 1}$ and $A \subseteq X$ be a [[Closed Set|closed]] subset. A [[Vector Field|vector field]] **along** $A$ is a [[Continuity|continuous]] map $F: X \to TM$ such that $\pi \circ X = \text{Id}_A$. $F$ is $C^{p - 1}$ **along** $A$ if for each $p \in A$, there is a [[Neighbourhood|neighbourhood]] $V \in \cn^o(p)$, on which $F$ has a $C^{p - 1}$ extension. > [!theorem] > > Let $X$ be a [[n-Manifold|n-manifold]] of class $C^p$, $A \subset X$ be a closed subset, and $U \supset A$ be an open set. If $F$ is a $C^{p - 1}$ vector field along $A$, then there exists a $C^{p - 1}$ extension of $F$ to $U$. > > *Proof*. For each $p \in A$, let $U_p \in \cn^o(p)$, assume without loss of generality that $U_p \subset U$, and $F_p$ be an extension of $f$ to $U$. Then $\bracs{U_p}_{p \in A}$ combined with $A^c$ is an open cover of $X$. Let $\bracs{\varphi_p}_{p \in A}$ combined with $\varphi_0$ be a [[Partition of Unity on n-Manifold|partition of unity]] subordinate to this cover. Define > $ > \ol{F} = \sum_{p \in A}\varphi_p F_p > $ > then since $\bracs{\supp{\varphi_p}}_{p \in A}$ is locally finite, for each $x \in X$ there exists $V \in \cn^o(x)$ such that $\ol{F}|_{V}$ is a finite sum of $C^{p - 1}$ functions, and is $C^{p - 1}$ at that point. As $\sum_{p}\varphi_pF_p = \sum_{p}\varphi_p F = F$ on $A$, $\ol{F}$ is an extension of $F$, and since $\supp{\sum_{p}\varphi_p} \subset U$, $\supp{\ol{F}} \subset U$.