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Is R mod 2pi a Compact Manifold?

  1. Jan 28, 2012 #1
    Hi,
    Why R mod 2pi is a Compact Manifold?
    Isn't this like a real line which is not compact?
    How should we prove it using a finite sub-cover for this manifold?
    bah
     
  2. jcsd
  3. Jan 28, 2012 #2

    morphism

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    R mod 2pi doesn't look like the real line; it looks like the interval [0,2pi] except the end points are identified. (So it's a circle...)
     
  4. Jan 28, 2012 #3

    mathwonk

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    consider the map from R to the circle defined by t-->(cos(t), sin(t)).
     
  5. Jan 28, 2012 #4

    lavinia

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    What is the topology that you want?
    If it is R with the topology of a line modulo the discrete group of integer multiples of 2pi, then use the definition of open set in the quotient topology to show that every open cover has a finite subcover. You need to know that a closed interval is compact.
     
  6. Jan 28, 2012 #5

    Bacle2

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    Isn't the quotient topology assumed when talking about a quotient?

    It seems like the open sets here would be the sets (a,b); 0<a,b<2∏ , which lift to

    the open sets U(a±2k∏, b± 2k∏ ); for k integer, in ℝ

    Then, since compactness is hereditary, and this topology is smaller than the subspace topology in [0,2∏],

    (which is a compact space), then the quotient is compact.

    EDIT:
    Correction: the quotient topology agrees with the subspace topology of S^1 in R^2. By, e.g., mathwonk's map f, which descends to

    a homeo f^ between R/~ and S^1, subspace (since the map is constant in R/~) , and which can be shown to be continuous and onto.

    Then, by homeo., R/~ is also compact. If you want an argument by subcovers, do the argument in S^1, and pull back to R/~ by the homeo f^
     
    Last edited: Jan 28, 2012
  7. Jan 28, 2012 #6

    Bacle2

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    Actually, to show the manifold part, there are results that the quotients of manifolds by some special types of group actions are themselves manifolds--you can, e.g., show that projective space is a manifold by using this result/theorem.
     
  8. Jan 28, 2012 #7
    Thanks guys,
    It helped me a lot. I did some studies on quotient topology and now I have a better understanding of the problem.
     
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