From https://pdfs.semanticscholar.org/e737/a4f8b93242910c050c2faf761236dcf60f64.pdf (Theorem 2.1) it follows:
(*) If $\pi:P\rightarrow M$ is a topological fibre bundle over a paracompat hausdorff topological space $M$ then it is a Hurewicz fibration and hence a Serre fibration. (In the category of topological spaces)
I found the analogous notion of homotopy lifting property and Serre/Hurewiz fibration in the category of smooth manifolds here When is a fibration a fiber bundle?.
My question is the following:
Does (*) holds in the category of smooth manifolds also?
I got an affirmative answer at least for Serre fibration in https://mathoverflow.net/questions/116231/given-a-serre-fibration-between-manifolds-how-ugly-can-it-be (where in the question it is mentioned "Clearly smooth fibre bundles are Serre fibrations" by @David Roberts.) But (no proof or any reference where such proof is discussed) is mentioned in the question.
So can anyone please give a proof of (*) or suggest any reference where such proof is discussed (at least for Serre fibration)?
Also it would be very helpful if someone can suggest some references or literature resources where the notion of Serre/Hurewicz fibration in the category of smooth manifolds is discussed.
Thanks in Advance.