TITLE: Is a map that is locally fiberwise equivalent to a product a Hurewicz fibration?
QUESTION [5 upvotes]: The following is a result I feel like I've seen some form of before, but can't figure out how to prove or find a reference for.  Suppose you have a map p:E \to B, with B paracompact, and suppose that every point in B has a neighborhood U such that there is a map p^{-1}(U) \to U \times F over U which is a fiber homotopy equivalence.  Does it follow that p is a Hurewicz fibration?  The converse, if B is locally contractible, is standard: a Hurewicz fibration is locally equivalent to a product.

REPLY [4 votes]: The answer is no; Allen Hatcher sent me the following:
An example where this fails is the projection of the letter L onto its horizontal base, which I'll call B.  The deformation retraction of L onto B is a fiberwise homotopy equivalence.  The homotopy lifting property fails:  Map a point to the left endpoint of B, then lift this to a point of L - B and take a homotopy that moves the left endpoint of B to the right endpoint.