Regarding fibrations between smooth manifolds

[aalma]

Definitions:
1. A map ##p : X → Y## of smooth manifolds is called a trivial fibration with fiber ##Z## which is also a smooth manifold, if there is a diffeomorphism ##θ : X → Y ×Z## such that ##p## is the composition of ##θ## with the natural projection ##pr_1:Y × Z → Y##.
2. A map ##p: X →Y## is a locally trivial fibration with fiber ##Z## if for all ##y \in Y## there exists an open neighborhood ##U⊂Y## of ##y## such that ##p: p^{−1}(U) → U## is a trivial fibration with fiber ##Z##.
3. A smooth map ##p : X → Y## is called a quotient map if the following conditions are fulfilled:
1. ##U ⊂Y## is open iff ##p^{−1}(U)## is open in X.
2. ##f : U →R## is smooth iff ##f ◦p## is a smooth function on ##p^{−1}(U)##.
Now, I need to show that any locally trivial fibration is a quotient map.

Let ##p:X\to Y## a locally trivial fibration.
Well I need to verify the two conditions for a quotient map

1. ##U ⊂ Y## is open iff ##p^{-1}(U)## is open in ##X##.
2. ##f : U → R## is smooth iff ##f ◦ p## is a smooth function on ##p^{-1}(U)##.

For the first condition, suppose that ##U## is an open subset of ##Y##. We want to show that ##p^{-1}(U)## is open in ##X##. By the definition of a locally trivial fibration, for every ##y \in U##, there exists an open neighborhood ##V_y## of ##y## such that ##p^{-1}(V_y)## is diffeomorphic to ##V_y × Z## via a diffeomorphism ##θ_y##. Since ##U## is an open subset of ##Y##, we can cover ##U## by the open neighborhoods ##V_y##. That is, ##U = ∪_y V_y##.
Now consider ##p^{-1}(U)##. For any point ##x \in p^{-1}(U)##, we have ##p(x) ∈ U##, so there exists a ##V_y## containing ##p(x)##. Since ##p^{-1}(V_y)## is diffeomorphic to ##V_y × Z## via ##θ_y##, we can write ##x = θ_y(y', z)## for some ##y' \in V_y## and ##z \in Z##. Moreover, since ##V_y## is an open neighborhood of ##y##, we can assume that ##y'## lies in a smaller open set ##U_y## contained in ##V_y##. Therefore, we have ##x = θ_y(y', z) ∈ U_y × Z##, which is contained in ##p^{-1}(V_y)##. Thus, we have shown that every point ##x \in p^{-1}(U)## is contained in an open set of the form ##U_y × Z##, which is diffeomorphic to an open set in ##Y × Z##. Therefore, ##p^{-1}(U)## is an open subset of ##X##.
Does this seem okay?
Next, let's consider the second condition. Suppose that ##f : U → R## is a smooth function on ##U##. We want to show that ##f ◦ p## is a smooth function on ##p^{-1}(U)##. For any point ##x \in p^{-1}(U)##, we have ##p(x) ∈ U##, so there exists a ##V_y## containing ##p(x)##. Since ##p^{-1}(V_y)## is diffeomorphic to ##V_y × Z## via ##θ_y##, we can write ##x = θ_y(y', z)## for some ##y' \in V_y## and ##z \in Z##. Moreover, since ##f## is smooth on ##U##, we can write ##f(y') = g_y(y')## for some smooth function ##g_y## on ##V_y##. Then we have:
##(f ◦ p)(x) = f(p(x)) = f(y') = g_y(y') = (g_y ◦ p')(y', z)##
where ##p'## is the projection ##Y × Z → Y##. We note that ##(g_y ◦ p')## is a smooth function on ##V_y × Z##, which is diffeomorphic to ##p^{-1}(V_y)## via ##θ_y##. Therefore, ##(g_y ◦ p')(y', z)## is a smooth function on ##p^{-1}(V_y)##, which contains ##x##. Since this is true for any ##V_y## containing p(x), we conclude that ##f ◦ p## is a smooth function on ##p^{-1}(U)##.
What do you think?Thanks in advance for any tips..

 

[mathwonk]

i apologize if this is unhelpful, but i hope it may be. First step is to realize it suffices to prove this for a trivial fibration p:YxZ-->Y, in which case it follows almost immediately from the definition of the product topology and the fact that p restricts to a diffeomorphism from Yx{q}-->Y, for any point q in Z.