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Given a covering space $p: E \to B$ over a connected space $B$, I want to show that it is a fibre bundle with a discrete fibre $F$.

More specifically, I want to know why, for every $x \in B$, $p^{-1}\{x\} \cong F$. That is, why does the fibre not change when $x$ does?

A possible way is to show that $x \mapsto |p^{-1}\{x\}|$ is a continuous function. But I can't prove this.

This is a related question, but doesn't answer my doubt.

Thanks.

Community
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Sayantan
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1 Answers1

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Hint: As you suggest, first show that $x\mapsto |p^{-1}(x)|$ is locally constant, and hence constant since $B$ is connected. To prove this, take a point $x_0\in B$ and choose a neighborhood $U$ of $x_0$ which is evenly covered by $p$. Can you show that $|p^{-1}(x)|$ is constant on $U$?

Once you have that, you can again use neighborhoods that are evenly covered to show $p$ is locally trivial.

Eric Wofsey
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