Corollary 1

Given any open covering $(A_i)_{i \in I}$ of a paracompact space $X$, there exists a continuous partition of unity $(f_i)_{i \in I}$ on $x$, subordinate to the covering $(A_i)$.

Let $(U_\lambda)_{\lambda \in L}$ be a locally finite open covering of $X$ which refines the covering $(A_i)_{i \in I}$; then there is a mapping $\varphi : L \to I$ such that $U_\lambda \subset A_{\varphi(\lambda)}$ for each $\lambda \in L$. By Propositions 3 and 4, there exists a continuous partition of unity $(g_\lambda)_{\lambda \in L}$ subordinate to $(U_\lambda)$. For each $i \in I$, put

this sum is defined and continuous since the supports of the $g_\lambda$ form a locally finite covering; moreover, the union $B_i$ of the supports of the $g_\lambda$ such that $\varphi(\lambda) = i$ is closed, by Proposition 4 of Chapter I, § 1, no. 6, and is contained in $A_i$. Since we have $f_i(x) = 0$ whenever $x \in \mathbf{C}B_i$, the support of $f_i$ is contained in $B_i$, and therefore in $A_i$. On the other hand, the family $B_i$ is locally finite, because for each $x \in X$ there is a neighbourhood $V$ of $x$ and a finite subset $H$ of $I$ such that $V \cap U_\lambda = \emptyset$ for all $\lambda \notin H$, and it follows therefore that $V \cap B_i = \emptyset$ for all $i \notin \varphi(H)$. Finally, for each $x \in X$ we have

and the proof is complete.