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Proper => Homeomorfismo

  1. Sep 18, 2011 #1

    Lie

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    Hello!

    F: X --> Y injection.

    It is true that if F is proper (the inverse image of any compact set is compact) then F: X --> F(X) is a homeomorphism?

    Thanks... :)
     
  2. jcsd
  3. Sep 18, 2011 #2

    quasar987

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    You need F to be continuous and Y to be Hausdorff and compactly generated. See Corolarry 4.97 of Lee's Introduction to topological manifolds.
     
  4. Sep 18, 2011 #3

    Lie

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    Yes, I had forgotten: F to be continuous and Y (X and) to be Hausdorff. :)

    Compactly generated = union of open compact ?

    Thanks... :)
     
  5. Sep 18, 2011 #4

    quasar987

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  6. Sep 19, 2011 #5

    Lie

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    Thanks!

    I showed that Y is locally compact space and therefore is compactly generated space.

    Grateful.
     
  7. Sep 19, 2011 #6

    quasar987

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    You're welcome. :)
     
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