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Thread: 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 $\displaystyle \beta (A)$ is the boundary of $\displaystyle A$.
    If $\displaystyle x$is a limit point of $\displaystyle \beta (A)$ and $\displaystyle \mathcal{O}$ is and open set such that $\displaystyle x\in\mathcal{O}$ implies $\displaystyle \left( {\exists y \in \mathcal{O}} \right)\left[ {y \in \beta (A)\backslash \{ x\} } \right]$.
    That means that because $\displaystyle y$ is a boundary point, $\displaystyle \mathcal{O} $ must contain a point of $\displaystyle A$ and a point not in $\displaystyle A$.
    How does this prove that $\displaystyle x$ is a boundary point of $\displaystyle A$? How does that show that $\displaystyle \beta (A)$ is closed?
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  3. #3
    MHF Contributor Drexel28's Avatar
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    $\displaystyle \partial E=\bar{E}\cap\overline{E'}$....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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