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Math Help - Proof of Topological Property

  1. #1
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    Proof of Topological Property

    Show that the boundary of any set D is itself a closed set.

    Thanks in advance for any advice...
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  2. #2
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    Quote Originally Posted by jzellt View Post
    Show that the boundary of any set D is itself a closed set.
    Suppose that \beta (A) is the boundary of A.
    If xis a limit point of \beta (A) and \mathcal{O} is and open set such that x\in\mathcal{O} implies  \left( {\exists y \in \mathcal{O}} \right)\left[ {y \in \beta (A)\backslash \{ x\} } \right].
    That means that because y is a boundary point,  \mathcal{O} must contain a point of A and a point not in A.
    How does this prove that x is a boundary point of A? How does that show that \beta (A) is closed?
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  3. #3
    MHF Contributor Drexel28's Avatar
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    \partial E=\bar{E}\cap\overline{E'}....sooo...
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  4. #4
    Member mabruka's Avatar
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    Maybe state what definition of boundary you are starting from ?
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