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Quick basic topology question.

  1. Dec 13, 2008 #1
    1. The problem statement, all variables and given/known data

    Is it true that if U is an open set, then U = Int(closure(U))?

    3. The attempt at a solution I feel like this may be true; I found counter-examples to the general form, Int(U)=(Int(closure(U)), but they all seem to hinge on U being not open (A subset of rationals in the reals, which is neither open nor closed).

    However I can't prove this one way or another; I'd like a nudging in the right direction of the proof is this is true, and just a, "it's false" if it's false, so I can keep hunting for a CE.
  2. jcsd
  3. Dec 13, 2008 #2


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    It's true. Where is the proof giving you a problem? You want to show if x is an element of U, then x is an element of int(closure(U)) and vice versa.
  4. Dec 13, 2008 #3
    lol, I think it being four in the morning was giving me trouble.

    So the first direction would be

    x in U -> x an interior point and x in Clos(U); Int(U) = all interior points in U(closure), which includes x, so x in Int(Clos(U))

    The other direction...

    x in Int(clos(U))-> x an interior point of Clos(U) = Int(U) by definition which = U, since U is open.

    Does that check out? The only reason I was having difficulty with this, is just intuitively, why is it not possible that closing the set/adding limit points will add points to the interior? I guess that means that if we had any interior point of U, it must have been originally contained in U if U is open. However, not if U isn't open. Why is this?
  5. Dec 13, 2008 #4


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    Ooops. Sorry. Actually, it is hard to prove because it's false for exactly the reason you pointed out. Take U to be the union of (0,1) and (1,2). My mistake.
  6. Dec 13, 2008 #5
    Ok, excellent; then I both understand why it is false and where that BS proof I just made fails.

    Thanks a ton.
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