Base space is connected then are all fibres isomorphic?

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I am studying about fibre bundles. If a fibre bundle is locally trivial then I understand the fibres are locally isomorphic. Also I understand that any compact Hausdorff base space cannot have a disjoint finite open cover for if I have a disjoint finite open cover then base space is disconnected. My question is, if my base space is connected, then are all the fibres isomorphic to each other?