Exercise 2, § 5

Let B be a topological space. Let $(X, q)$ be a locally trivial B-fiber space and let $(E, p)$ be a principal fiber space with base B and group G.

a) Suppose that there exists a trivialization $\mu: E\times F\rightarrow E\times_BX$ of the locally trivial E-fiber space $E\times_BX$ which is compatible with the right operations of G defined by $(x, f)\cdot g= (x\cdot g, g^{-1}\cdot f)$ and $(x, y)g= (x\cdot g, y)$. Prove that the fiber space X is associable with E.

b) More precisely, construct a bijection between the set of these trivializations and the set of B-isomorphisms of $E\times^GF$ onto X.