Remark 1

Let $(E, p)$ be a locally trivial fibred B-space and let A be a subset of B. The A-space $E_A= (\overset{-1}{p}(A), p_A)$ deduced from E by passing to the subspaces is a locally trivial fibre space. If E is trivializable, the same is true of $E_A$.

In fact, for every point $a$ of A, there exist an open neighbourhood U of $a$ in B, a topological space F and a U-isomorphism $g:\overset{-1}{p}(U)\rightarrow$ $U\times F$. The mapping $f$ induces an $(A\cap U)$-isomorphism of $p^{-1}_A(A\cap U)$ onto $(A\cap U)\times F$, which proves that $E_A$ is a locally trivial fibred A-space.