Cylinder as Fibre bundles

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I have to show that the cylinder C is a fibre bundle over $S^1$ with fibre an open interval and I have to write a trivialization and the cocycles. I think that this is a trivial bundle, because I can see it as $S^1 \times (a,b)$ (how I can argue this better?). In this trivial case, a trivilization is given only by the identity and the cocycle is trivially the identity too. Is what i wrote right? Thanks.