Proposition 18

The sum (§ 2, no. 4, Example 3) of a family $(X_i)_{i \in I}$ of paracompact spaces is paracompact.

Let $X$ be the sum of the $X_i$, and let $(V_\lambda)_{\lambda \in L}$ be an open covering of $X$. The covering formed by the open sets $X_i \cap Y_\lambda$ is finer than $(V_\lambda)$. For each $i \in I$, let $(U_{i,\mu})_{\mu \in M_i}$ be a locally finite open refinement of the covering $(V_\lambda \cap X_i)_{\lambda \in L}$; then the open covering of $X$ formed by the

is locally finite and refines the original covering $(V_\lambda)$.