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Math Help - interior points question

  1. #1
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    interior points question

    Let A and B be subsets of R^n with A^0, B^0 denoting the sets of interior points for A and B respectively. Prove that A^0UB^0 is a subset of the interior of AUB. Give an example where the inclusion is strict.
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  2. #2
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    One reason we ask people to show there work is that there are a number of different ways of defining "interior points" and without seeing what you are trying we don't know which is appropriate for you. I suspect you are using "p is an interior point of p if and only if there exist a neighborhood of p that is a subset of A".

    So if p is in A^O\cup B^O then it is in both A^O and B^O. Since it is in A^O there exist some neighborhood of p, N_{\delta_1}(p) ( \delta_1 is the radius), such that N_{\delta_1}(p)\subset A. Since p is in B^O, there exist some neighborhood of p, N_{\delta_2}(p) such that N_{\delta_2}(p)\subset B.

    Can you finish now?
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  3. #3
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    @HallsofIvy
    If p\in A^0 \cup B^0 \Rightarrow p\in A^0 \text{ or } p\in B^0. It is not necessary for p\in A^0 \text{ and } p\in B^0 that case would be necessary for the intersection not union.
    Last edited by lvleph; September 15th 2010 at 05:14 AM.
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    Quote Originally Posted by wopashui View Post
    Let A and B be subsets of R^n with A^0, B^0 denoting the sets of interior points for A and B respectively. Prove that A^0UB^0 is a subset of the interior of AUB. Give an example where the inclusion is strict.
    \forall p\in A^o\cup B^o\Rightarrow p\in A^o\, or \, p\in B^o \Rightarrow \exists \delta>0 \,such\, that\, N_\delta(p)\subset A \,or\, N_\delta(p)\subset  B\Rightarrow \Rightarrow\exists \delta>0 \,such\, that\, N_\delta(p)\subset A\cup B \Rightarrow p \in (A\cup B)^o.

    Short version \forall p\in A^o\cup B^o \Rightarrow p\in (A\cup B)^o, hence A^o\cup B^o\subseteq (A\cup B)^o.
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  5. #5
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    Quote Originally Posted by lvleph View Post
    @HallsofIvy
    If p\in A^0 \cup B^0 \Rightarrow p\in A^0 \text{ or } p\in B^0. It is not necessary for p\in A^0 \text{ and } p\in B^0 that case would be necessary for the intersection not union.
    Absolutely! For some reason I was thinking "intersection". If p\in A^O\cup B^O then p is in either A^O or B^O so either there exist neighborhood N_{\delta}(p)\in A^O or there exist N_{\delta}(p)\in B^O. Fortunately, either of those is enough to conclude that N_{\delta}(p)\in A^O\cup B^O, as mathoman shows.
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