Local trivialization

Remarks

Example: let $E = \R \times S^1$, $B = S^1$ and $\pi(s, z) = z$. Then $\varphi(s, z) = (z, s)$ over $U = S^1$ is a local trivialization with fiber $\R$, since it is a homeomorphism and its first coordinate is $\pi(s, z) = z$.

Non-example: for the double cover $\pi \colon S^1 \to S^1$, $\pi(z) = z^2$, there is no local trivialization over $U = S^1$ with fiber a two-point space, since $\pi^{-1}(S^1) = S^1$ is connected and $S^1 \times F$ is not. Over each proper open arc, however, such a trivialization exists.

The map $\pi$ is a fiber bundle with fiber $F$ exactly when the open sets carrying a local trivialization cover $B$. If $\varphi_U$ and $\varphi_V$ are local trivializations, then $\varphi_U \circ \varphi_V^{-1}$ on $(U \cap V) \times F$ has the form $(b, f) \mapsto (b, g_{UV}(b)(f))$, and the maps $g_{UV}$ are the transition functions.