Lemma 1

*Let $X$ be a locally compact space, $K$ a compact subset of $X$, and $(V_k)_{1 \leq k \leq n}$ a finite covering of $K$ by open sets of $X$. Then, there exist $n$ continuous mappings $f_k$ of $X$ into $[0, 1]$, such that the support of $f_k$ is contained in $V_k$ for $1 \leq k \leq n$ and such that $\sum_{k=1}^n f_k(x) \leq 1$ for all $x \in X$ and $\sum_{k=1}^n f_k(x) = 1$ for all $x \in K$.

For, let $X'$ be the compact space obtained by adjoining to $X$ a point at infinity $\omega$ (GT, I, §9, No. 8, Th. 4); the sets $V_0 = X' - K$ and $V_k$ ($1 \leq k \leq n$) form an open covering of $X'$. Let $(f_k)_{0 \leq k \leq n}$ be a continuous partition of unity subordinate to this covering of $X'$ (GT, IX, §4, No. 3, Prop. 3); the functions $f_k$ with index $k \geq 1$ satisfy the conditions of the lemma.