I know that this question has already been asked, but I couldn't find a clear answer.
I have to show that the cylinder and the Möbius strip are fiber bundles over $S^1$ with fiber an open interval and I have to write down explicitly the transition functions and the cocycles.
Cylinder.
The cylinder is $E:=I\times S^1$ where $I$ is our open interval.
Let's consider $$ \pi : I \times S^1 \to S^1 $$ the projection on the second factor (by definition it's continuous and surjective).
Let's take $S^1$ as basis of the fiber bundle and let $\{U_\alpha\}_\alpha$ be an open covering of $S^1$.
$\pi_1:U_\alpha\times I\to U_\alpha$ is the projection on the first factor.
The trivialization is $$ \phi_\alpha:\pi^{-1}(U_\alpha)\to U_\alpha\times I\\ \phi_\alpha:=id|_{U_\alpha\times I} $$ In this case the transition functions are such that for all $\alpha,\beta$ $$ \phi_{\alpha\beta}=\phi_\alpha\circ\phi_\beta^{-1}=id|_{(U_\alpha\cap U_\beta)\times I}. $$ Hence the cocycles are just $$ g_{\alpha\beta}:U_\alpha\cap U_\beta\to Aut(I)\\ p\mapsto id|_I $$
Möbius band Let's start with the closed interval $[0,1]$ and the map $$ p:[0,1]\to S^1\\ x\mapsto e^{i2\pi x} $$ which identifies the point $0$ with the point $1$.
Let's consider $[0,1]\times I$, where $I$ is an open interval, let's say $(0,1)$. Let's define an equivalence relation on $[0,1]\times I$: $$ (0,y)\sim (1,1-y) $$ and we obtain the Möbius band $E$.
Let's consider an open covering $\{U_\alpha\}_\alpha$ of sphere $S^1$ and let $$ \pi:E\to S^1\\ [(x,y)]\mapsto p(x) $$ be continuous and surjective and $$ \pi_1:U_\alpha\times I\to U_\alpha $$ be continuous and surjective.
I can't find a trivialization. I just know that I have to choose a proper open covering of $S^1$ like two open interval overlapped.