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Math Help - Is it an open set?

  1. #1
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    Is it an open set?

    If O is open and F is closed. Is O/F open?
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  2. #2
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    Quote Originally Posted by Cerny View Post
    If O is open and F is closed. Is O/F open?

    What do you mean by O/F??

    Tonio
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    x is in O/F iff x is in O but not in F.

    I understand this should be a backslash. I put the blame on my keyboard...
    Last edited by Cerny; September 18th 2010 at 09:32 PM.
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    If there is no such x, then O/F is an empty set. Hence O/F is open. I couldn't show that the result holds when O/F is nonempty.
    Last edited by Cerny; September 18th 2010 at 07:48 PM.
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  5. #5
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    Hello,

    By definition of the set difference, O\setminus F=O\cap F^c
    But by definition of a closed set in a topology, F^c is an open set.
    And from one of the axioms of a topology, the intersection of two open sets is still an open set.
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