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Ext(ExtA) = intA ?

  1. Mar 10, 2008 #1
    Definition: Ext(A) = int (X-A) = X - Cl(A)

    Is it true that Ext(ExtA) = intA ?
     
  2. jcsd
  3. Mar 10, 2008 #2

    Dick

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    Yes. Can you prove it?
     
  4. Mar 10, 2008 #3
    So far, I've only proven that Ext(ExtA) c X-extA. I can't arrive at Ext(ExtA) c intA.
    Are you sure the statement is true?
     
    Last edited: Mar 11, 2008
  5. Mar 11, 2008 #4

    Dick

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    If x is contained in Int(A) then there is an open neighborhood of x that is not in Ext(A). Can you show that makes it a member of Ext(Ext(A))?
     
  6. Mar 11, 2008 #5
    X=R, A = rationals.

    Then ext(extA)=ext(int(irrationals))= ext (empty) = R

    But int(A) = empty. The statement is false.
     
    Last edited: Mar 11, 2008
  7. Mar 11, 2008 #6

    Dick

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    Ext(rationals)=empty. Ext(empty)=R. Int(rationals)=empty. Yep. Good point. Not trusting authority is a good thing. Keep it up. Sorry, not thinking.
     
  8. Mar 11, 2008 #7
    I thought it was true too when I pictured A being a circle in the xy-plane. I was wondering why it was so hard to prove it.
     
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