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Math Help - Question about open and closed sets

  1. #1
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    Question about open and closed sets

    Hi guys,

    I have two questions wanting to clarify.

    1) My lecture notes say that if a set is open then its complement is closed. However some books also say that the converse is true: that is if a set is closed, then its complement is open. Is this correct?

    2) Is a boundary point a limit point? The definition says that a boundary point x which belongs to S is a point if, for all epsilon > 0, the neighborhood of x contains both S and the complement of S. Doesn't this make x a limit point also? Since you can approach x from elements in the set S.

    Thanks for reading.
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  2. #2
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    Quote Originally Posted by sakodo View Post
    1) My lecture notes say that if a set is open then its complement is closed. However some books also say that the converse is true: that is if a set is closed, then its complement is open. Is this correct?

    2) Is a boundary point a limit point? The definition says that a boundary point x which belongs to S is a point if, for all epsilon > 0, the neighborhood of x contains both S and the complement of S. Doesn't this make x a limit point also? Since you can approach x from elements in the set S.
    Your #1 is correct. If \mathcal{O} is open then \mathcal{O}^c is closed.
    If \mathcal{C} is closed then \mathcal{C}^c is open.

    For #2 consider \mathcal{A}=\{0\}\cup [1,2].
    Then 0 is a boundary point of \mathcal{A} but not a limit point.
    Last edited by Plato; March 14th 2011 at 12:48 PM. Reason: correction
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  3. #3
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    Quote Originally Posted by sakodo View Post
    Hi guys,

    I have two questions wanting to clarify.

    1) My lecture notes say that if a set is open then its complement is closed. However some books also say that the converse is true: that is if a set is closed, then its complement is open. Is this correct?
    Indeed, in most texts, a "closed" set is defined as "the complement of an open set" and then it is proved that the complement of a closed set is an open set.

    2) Is a boundary point a limit point? The definition says that a boundary point x which belongs to S is a point if, for all epsilon > 0, the neighborhood of x contains both S and the complement of S. Doesn't this make x a limit point also? Since you can approach x from elements in the set S.

    Thanks for reading.
    The definition of "boundary point" is, as you say, that every neighborhood contains points of S and of the complement of S. The definition of "limit point" is that every neighborhood contains points of S other than X itelf. Plato's example gives a point, 0, such that every neighborhood of 0 contains only 0 from the set. It is a boundary point but not a limit point. Such points are called "isolated points" of S.
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  4. #4
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    Quote Originally Posted by Plato View Post
    Your #1 is correct. If \mathcal{O} is open then \mathcal{O}^c is closed.
    If \mathcal{C} is closed then \mathcal{C}^c is closed.
    Plato meant "open" as the last word here.

    For #2 consider \mathcal{A}=\{0\}\cup [1,2].
    Then 0 is a boundary point of \mathcal{A} but not a limit point.
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