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Thread: interior points and proof of subsets

  1. #1
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    Exclamation interior points and proof of subsets

    Let A be a subset of R^d. and
    set E=A Union (non interior points of A complement).

    a) prove E is a closed subset of R^d

    b) With A and E as above, let F be a closed subset of R^d, and suppose that A is a subset of F. prove that E is a suset of F.
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  2. #2
    Senior Member roninpro's Avatar
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    Hello.

    I'd like to help you with the first part.

    One way to show that a set is closed is to show that it contains all of its limit points. So, take $\displaystyle A^c$ as in the problem, and put $\displaystyle E=\bigcup \{x\}$ where $\displaystyle x$ is a noninterior point of $\displaystyle A^c$.

    Let $\displaystyle y$ be a limit point of $\displaystyle E$. Then, there is a sequence of points from $\displaystyle E$ that converges to $\displaystyle y$. If we can show that $\displaystyle y$ is an element of $\displaystyle E$ (i.e. a noninterior point of $\displaystyle A^c$), then $\displaystyle E$ is closed and we are done. To prove this, you should think about what happens if $\displaystyle y$ is not an element of $\displaystyle E$; in other words, what happens when $\displaystyle y$ is in the interior of $\displaystyle A^c$?

    Good luck.
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