Lemma

Let E be a separated topological space. Suppose that E has a locally finite open covering $(V_i)_{i\in I}$ such that, for every $i\in I,\overline{V_i}$ is a paracompact space. Then the space E is paracompact.

Let $(W_j)_{j\in J}$ be an open covering of E; we shall prove that there exists an open locally finite covering $(A_k)_{k\in K}$ of E which is finer than the covering $(W_j)_{j\in J}$. For each $i\in I$, let $(A'_{\ell})_{\ell\in K_i}$ be a locally finite covering of $\overline{V_i}$ by open sets of $\overline{V_i}$, finer than the covering $(W_j\cap \overline{V_i})_{j\in J}$. Let K be the sum of the family $(K_i)_{i\in I}$ (E, II, p. 30, Def. 8). For each element $k= (\ell , i)$ of K, put $A_k= A'_{\ell}\cap V_i$. Then $A_k$ is open in E and one has $\bigcup_{k\in K}A_k=\bigcup_{i\in I}V_i= E$; moreover, for each $k\in K$, there exists an index $j\in J$ such that $A_k\subset W_j$. Thus, the family $(A_k)_{k\in K}$ is an open covering of E finer than $(W_j)_{j\in J}$. It remains to prove that the family $(A_k)_{k\in K}$ is locally finite. Let $x\in E$; there exists an open neighbourhood U of $x$ which meets $V_i$ only for $i$ belonging to a finite subset $I'$ of I. For each $i\in I',x$ has an open neighbourhood $U_i\subset U$ which meets only a finite number of the open sets $A_k$, for $k\in K_i\times \{i\}:$ this is obvious if $x$ does not belong to $\overline{V_i}$, and if $x$ belongs to $\overline{V_i}$, this follows from the property of local finiteness of the covering $(A'_{\ell})_{\ell\in K_i}$ of $\overline{V_i}$. It follows that $V'=\bigcap_{i\in I'}U_i$ is an open neighbourhood of $x$ which meets only a finite number of the $A_k,k\in K$, and the covering $(A_k)_{k\in K}$ is locally finite.

Let us prove the proposition. Let E be a covering of B, let $p$ denote its projection, and suppose that the space B is paracompact. Let $(A_i)_{i\in I}$ be an open locally finite covering of B such that, for each $i\in I$, the covering E is trivializable over $A_i$. For each $i\in I$, let $F_i$ be a discrete topological space and let $g_i:\overset{-1}{p}(A_i)\rightarrow A_i\times F_i$ be a trivialization of E over $A_i$. Let $(B_i)_{i\in I}$ be an open covering of B such that, for each $i\in I$, one has $\overline{B_i}\subset A_i$ (TG, IX, p. 49, Prop. 4 and p. 48, Cor. 1). For each $i\in I$, put $V_i=\overset{-1}{p}(B_i)$; one has $\overline{V_i}\subset \overset{-1}{p}(B_i)\subset \overset{-1}{p}(A_i)$ and $V_i=\overset{-1}{g_{i}}(B_i\times F_i)$, whence $\overline{V_i}\subset \overset{-1}{g_{i}}(B_i\times F_i)$. Since B is paracompact, $\overline{B_i}$ is paracompact (TG, I, p. 69, Prop. 16) and $\overline{B_i}\times F_i$ is paracompact (TG, I, p. 70, Prop. 18). It follows that $\overset{-1}{g_{i}}(B_i\times F_i)$ is paracompact, hence so is $\overline{V_i}$ (TG, I, p. 69, Prop. 16). The family $(V_i)_{i\in I}$ is, by construction, an open locally finite covering of E. Finally, the space E is separated (I, p. 26, Remark 3). It therefore satisfies the hypotheses of the lemma, whence the proposition.