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Math Help - Interior points

  1. #16
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    You seem to be having trouble with the basic definition of what it means to be an interior point.

    Since p is an interior point of A, there is an open set O\subset A with p\in O. Similarly there is an open set U\subseteq B with p\in U. Then O\cap U\subseteq A\cap B is an open set with p\in O\cap U. This shows that p\in (A\cap B)^o.
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  2. #17
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    @ alice8675309
    Here is a way to think about both the interior of a set and the closure of a set.
    The interior of a set is the ‘largest’ open subset of the set.
    The closure of a set is the ‘smallest closed set containing the set as a subset.

    Here is another suggestion. If you can find the small book Elementary Theory of Metric Spaces by Robert B Reisel, then that would be a great introduction for you. It is written as a self-study or Moore-style class.
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  3. #18
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    Thanks for the help guys and Plato, thanks for suggesting the book. I'm obviously struggling with this stuff, but I HAVE to take it. So i appreciate all the patients and helpful hints/answers and i apologize for posting so much. Thanks
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  4. #19
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    No need to apologize. Just keep trying things and posting your attempted solutions. Eventually after working hard to prove these things on your own things will click.
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  5. #20
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    Quote Originally Posted by alice8675309 View Post
    What would the formal proof look like for this problem?

    Prove that the interior of an intersection is the intersection of the interiors:

    (A∩B) =A∩B
    Any point p in A^B must be a common point of A and B. There are two possibilities:
    p is an interior point of both A and B, or
    p is a boundary point of A and or B.

    If p is an interior point of A & B, there must be a neighborhood of p containing only points common to A and B, and this neighborhood becomes the neighborhood of p in A^B and so is an interior point of A^B.

    If p is a boundary point of, say, A, every neighborhood of p will contain points in A and not in A and so will have some points in common and not in common with the same point in B. Thus p is a boundary point of A^B.

    Therefore the only interior points of A^B will be common interior points of A & B: Ao^Bo.

    I cannot give a simpler explanation.
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