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A question from Munkres:

Let $p:E\rightarrow B$ be a covering map; let $B$ be connected. Show that if $p^{-1}(b_0)$ for some $b_0\in B$ has $k$ elements, then $p^{-1}(b)$ has $k$ elements for each $b\in B$.

If I were to show that $p^{-1}(b)$ is bijective to $p^{-1}(b_0)$ then I would be done with the proof. But I don't see how, I mean all I know is that for each $b\in B$ there's a nbhd $U_b$ that is evenly covered by $p$; I think I know that $B \subset \bigcup_{b\ne b_0 , b\in B}\cup U_{b_0}$, but don't see how does this help?

Any hints? Thanks.

MathematicalPhysicist
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  • try to use the fact that given any path, there exists a unique liftibg inside the covering space with a fixed initial point – Anubhav Mukherjee Sep 24 '15 at 17:38
  • @Anubhav.K the definition of lifting in Munkres appears in a section after this question first appears so I assume that there's a solution without the use of lifting; either way I don't understand your approach, can you elaborate on your approach? thanks. – MathematicalPhysicist Sep 24 '15 at 17:53
  • in that case you can define a map $f : B \to \mathbb{N}$ s.t $b \mapsto |p^{-1}(b)|$, and try to prove that this map is constant by proving it is actually continuous – Anubhav Mukherjee Sep 24 '15 at 18:02
  • @Anubhav.K What topology does $\mathbb{N}$ have? I mean in order to show that $f$ is continuous I need to show that if $V$ is open in $\mathbb{N}$ then $f^{-1}(V)$ is open in $B$; should I use the cofinite topology or another topology? – MathematicalPhysicist Sep 24 '15 at 18:11
  • $\mathbb{N}$ is the set of positive intiger, topology should be induced topology, and that is actually discrete topology, think – Anubhav Mukherjee Sep 24 '15 at 18:20
  • Ok, then I take an open set $V$ from $\mathbb{P}(\mathbb{N})$, how do I use the fact $p$ is a covering map? I also don't see how to use the fact that $B$ is connected. – MathematicalPhysicist Sep 24 '15 at 18:57
  • The point is that locally around $b$ there $k$-elements in each fiber, so if you keep covering by open sets and these open sets intersects, then in each of these ones there will be $k$-elements in each fiber. To formalize this let $S$ be the set of points such that the fiber have $k$-elements, prove that this set is clopen. – user40276 Sep 24 '15 at 18:59

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The usual elegant way to solve this is to define a locally constant function and noting that such functions are constant if the domain is connected. Check out the candidate defined on $B$ mapping $ x\mapsto \#p^{-1}x$ the number of elements in its preimage.

Daniel Valenzuela
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