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Math Help - 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 A^c as in the problem, and put E=\bigcup \{x\} where x is a noninterior point of A^c.

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

    Good luck.
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