The following assertion appears in Milnor's Characteristic Classes.
The restrictions of a vector bundle $E \to X×I$ over $X \times \{0\}$ and $X \times \{1\}$ are isomorphic if $X$ is paracompact.
This seems like a strange assertion to me. Could anybody perhaps sketch with their own motivation as to why this is true?
Let $i_t:X\rightarrow X\times I$ defined by $i_t(x)=(x,t)$. Consider $f_t:E_t\rightarrow X$ the pullback of $E\rightarrow X\times I$ by $i_t$. Suppose that $E$ is a $n$-vector bundle. We denote by $g_t$ the principal $Gl(n,R)$ vector bundle associated to $f_t$ and by $g$ the $Gl(n,R)$ vector bundle associated to $E\rightarrow X\times I$. Since $X$ is paracompact, $g$ is the pullback of the universal bundle $EGl(n,R)\rightarrow BGl(n,R)$ by the classifying map $u:X\times I\rightarrow BGl(n,R)$. The bundle $g_t$ is the pullback of the universal bundle $EGl(n,R)\rightarrow BGl(n,R)$ by $u\circ i_t$. So we have an homotopy between $u\circ i_0$ and $u\circ i_1$. This implies that $g_0$ is isomorphic to $g_1$ and henceforth $f_0$ is isomorphic to $f_1$.