Sufficient condition for weak homotopy equivalence

Question

I am looking for a proof of the following theorem.

Let $p: E \to B$ a continuous map. Suppose that $B$ has an open cover $O$ closed under finite intersections and such that for each $U \in O$ the restriction $p|_{p^{-1}(U)}:p^{-1}(U) \to U$ is a weak homotopy equivalence. Then $p$ is a weak homotopy equivalence.

Answer 1

You can find a slightly stronger version of this theorem as Corollary 4K.2 in Hatcher's Algebraic Topology or Theorem 6.7.11 in tom Dieck's Algebraic Topology, which says the following:

Let $p:E\to B$ be a continuous map. Suppose $B$ has an open cover $(U_i)_{i\in I}$ and $E$ has an open cover $(V_i)_{i\in I}$ such that $p(V_i)\subseteq U_i$ for each $i$ and the restriction of $p$ to $\bigcap_{i\in F} V_i\to \bigcap_{i\in F} U_i$ is a weak homotopy equivalence for each nonempty finite subset $F\subseteq I$. Then $p$ is a weak homotopy equivalence.

To deduce your version from this, let $I=O$, $U_i=i$, and $V_i=p^{-1}(U_i)$. Since $O$ is closed under finite intersections, $\bigcap_{i\in F} U_i$ will always just be some element $U\in O$ and then $\bigcap_{i\in F} V_i=p^{-1}(U)$ so $\bigcap_{i\in F} V_i\to \bigcap_{i\in F} U_i$ is indeed a weak homotopy equivalence.