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If p is a covering map with B compact and fiber of b finite, E compact

  1. Apr 16, 2014 #1
    1. The problem statement, all variables and given/known data
    Let [itex] p: E \rightarrow B [/itex] be a covering map.

    If B is compact and[itex]p^{-1}(b)[/itex] is finite for each b in B, then E compact.

    Note: This is a problem from Munkres pg 341, question 6b in section 54.

    3. The attempt at a solution

    I begin with a cover of E denote it [itex]\{U_\alpha\}[/itex].

    I want to reduce this to a finite subcover (thus showing that E is compact).

    First I use the fact that p is a covering map and thus open to send this cover of E to a cover of B.

    Denote the image of [itex]\{U_\alpha\}[/itex] under p by [itex]\{W_\alpha\}[/itex]

    Then since B is compact I can reduce this to a finite subcover: [itex]\cup_{i=1}^n W_i[/itex].

    Here is where I get stuck, I'm not sure how to send this finite subcover of B back over to E. I'm not even sure if I'm going about this the right way.

    Any help is greatly appreciated, thanks.
     
    Last edited: Apr 16, 2014
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  3. Apr 16, 2014 #2

    micromass

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    The natural thing to do is of course looking at ##p^{-1}(W_i)##. The problem is that

    [tex]U_i\subseteq p^{-1}(W_i)[/tex]

    and equality does not hold in general because ##p## is not injective. So maybe you can do something with the fact that ##p## is a local homeomorphism and that it has finite fibers.

    One thing I would do is first to replace the ##U_i## by a cover of smaller sets ##V_j## such that ##p:V_j\rightarrow p(V_j)## is a homeomorphism between open sets. Then think about how you can write ##p^{-1}(p(V_j))##.
     
  4. Apr 16, 2014 #3
    Why are we allowed to replace the [itex] U_\alpha [/itex] by a smaller cover? But then once we have that it's just as simple as saying we can reduce the [itex]V_j[/itex] to a finite subcover since after we map them to B we can take a finite subcover of them and map them back using [itex] p^-1[/itex] since each open set in the image is then homeomorphic to its pre-image?
     
    Last edited: Apr 16, 2014
  5. Apr 16, 2014 #4

    micromass

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    First try to obtain a finite cover of the ##V_j##. Then you can use that ##V_j\subseteq U_\alpha## for some ##\alpha## to obtain a finite cover of the ##U_\alpha##.

    By the way, in topology we call the ##V_j## a refinement of the ##U_\alpha##. The fact that the ##V_j## has a finite subcover can then be stated as "Every open cover has a finite open refinement". This statement is equivalent to compactness.

    Are you using that ##p^{-1}(p(V_j)) = V_j##? This is still not true. It is only true for injective maps. However, because you are dealing with a covering map, you can write ##p^{-1}(p(V_j))## in a better and more convenient form.
     
  6. Apr 16, 2014 #5
    Sorry, I'm still confused about why we're even allowed to replace [itex]U_\alpha[/itex] by the [itex]V_j[/itex] at all.

    We can write [itex]p^{-1}(p(V_j))[/itex] as a disjoint union of open sets in E, each of which maps homeomorphically to [itex]p(V_j)[/itex]. But this must be finite because of the finiteness of the fibers?
     
  7. Apr 16, 2014 #6

    micromass

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    Yes, that is something you should think about.

    Yes, even better: we can write ##p^{-1}(p(V_j))## as the finite union of some ##V_i##. So there exist a finite set ##I## such that

    [tex]p^{-1}(p(V_j)) = \bigcup_{i\in I} V_i[/tex]

    Using this, can you then find a open cover of the ##V_j## and hence of the ##U_\alpha##?
     
  8. Apr 16, 2014 #7
    I'm honestly so lost.
     
  9. Apr 16, 2014 #8

    micromass

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    You've done most of the proof! Where are you lost?
     
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