Lemma 3

Let B be a topological space and let $(E, p)$ be a locally trivial $B\times \mathbf{I}$-fibered space. Every point $a$ of B possesses a neighbourhood V such that the $V\times \mathbf{I}$-space $E_{V\times\mathbf{I}}$ is trivializable.

Let $a$ be a point of B; for every point $t$ of $\mathbf{I}$, there exists an open neighbourhood $W_t$ of $t$ in $\mathbf{I}$ and a neighbourhood $V_t$ of $a$ in B such that E is trivializable over $V_t\times W_t$. There then exist an integer $n >0$ and, for every integer $i$ such that 1 $\leqslant i\leqslant n$, a point $t_i$ of $\mathbf{I}$ such that the interval $[\frac{i-1}{n},\frac{i}{n}]$ is contained in $W_{t_i}($III, p. 272, lemma 4). Put $V =\cap_{1\leqslant i\leqslant n}V_{t_i}$. The fibered space E is trivializable over $V\times$ $[\frac{i-1}{n},\frac{i}{n}]$ for every integer $i$ such that $1\leqslant i\leqslant n$. Lemma 3 then follows from Lemma 2 by induction on $n$.

Let us now prove the proposition. By lemma 3, there exists an open covering $(U_j)_{j\in J}$ of B such that, for every $j\in J$, E is trivializable over $U_j\times \mathbf{I}$. Since the space B is paracompact, we may suppose the covering $(U_j)_{j\in J}$ locally finite (TG, IX, p. 49) and choose a covering $(A_j)_{j\in J}$ of B where, for every $j\in J$, the set $A_j$ is closed in B and contained in $U_j$ (TG, IX, p. 49, prop. 4 and p. 48, cor. 1).

For every open subset U of B, let us denote by $\mathscr{F}(U)$ the set of $U\times \mathbf{I}$-isomorphisms of $\overset{-1}{p}(U\times \mathbf{I})$ onto $\overset{-1}{p_{1}}(U)\times \mathbf{I}$ which induce the identical mapping of $\overset{-1}{p}(U\times \{1\})$ onto $\overset{-1}{p_{1}}(U)\times \{1\}$. For every pair $(V,U)$ of open sets of B such that $U\subset V$, let us denote by $r_{UV}:\mathscr{F}(V)\rightarrow \mathscr{F}(U)$ the mapping which, to a $(V\times \mathbf{I})$-isomorphism $g:\overset{-1}{p}(V\times \mathbf{I})\rightarrow \overset{-1}{p_{0}}(V)\times \mathbf{I}$, associates the $(U\times \mathbf{I})$-isomorphism deduced from $g$ by passing to the subspaces. The pair $\mathscr{F}= ((\mathscr{F}(U)),(r_{UV}))$ is a sheaf on B (I, p. 45, example 4). To prove the proposition, it is enough to prove that the sheaf $\mathscr{F}$ is soft (I, p. 64).

Let $j\in J$, let A be a closed subset of $A_j$, let V be an open set in B such that $A\subset V\subset U_j$, and let $g$ be an element of $\mathscr{F}(V)$. There exists an open neighbourhood W of A such that $\overline{W}\subset V$, for a paracompact space is normal (TG, IX, p. 49, prop. 4). We shall prove that there exists an element $g'$ of $\mathscr{F}(U_j)$ such that $g'|W =g|W$. Corollary 2 (I, p. 65) of prop. 6 then implies that the sheaf $\mathscr{F}$ is soft.

Since the $B\times \mathbf{I}$-space E is trivializable over $U_j\times \mathbf{I}$, we may suppose that $\overset{-1}{p}(U_j\times \mathbf{I}) = U_j\times \mathbf{I}\times F$, where F is a topological space. The element $g$ of $\mathscr{F}(V)$ is then a $V\times \mathbf{I}$-isomorphism of $V\times \mathbf{I}\times F$ onto itself which induces the identical mapping of $V\times$ $\{1\}\times F$. Apply cor. 5 of IV, p. 439, to the spaces $X'= X$, $X = U_j$, A = W, U = V, and Y = Z = F, to the mapping $g: V\times \mathbf{I}\times F\rightarrow V\times \mathbf{I}\times F$ and to the identical mapping of $U_j\times \{1\} \times F$. There therefore exists a $U_j\times \mathbf{I}$-isomorphism $g'$ of $U_j\times \mathbf{I}\times F$ onto itself which induces the identical mapping of $U_j\times \{1\} \times F$ and which coincides with $g$ on $\overline{W}\times \mathbf{I}\times F$, and a fortiori on $W\times \mathbf{I}\times F$. Hence the proposition.