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Math Help - Prove that A^0 U B^0 is a subset of the interior of A U B

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    Prove that A^0 U B^0 is a subset of the interior of A U B

    Let A and B be subsets of \Re^n with A^0, B^0 denoting the sets of interior points for A and B respectively. Prove that A^0\cup B^0 is a subset of the interior of A\cup B. Give an example where the inclusion is strict.
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    Quote Originally Posted by Runty View Post
    Let A and B be subsets of
    \Re^n with A^0, B^0 denoting the sets of interior points for A and B respectively. Prove that A^0\cup B^0 is a subset of the interior of A\cup B. Give an example where the inclusion is strict.
    Recall that the interior of a set is the largest open subset of a set.
    The union of two open sets is open. So what about A^o\cup B^o?
    Is that union a open subset of A\cup B?

    In \mathbb{R}^1 let A=[0,1]~\&~B=[1,2] what are A^o\cup B^o~\&~(A\cup B)^o?
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