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Math Help - interior points

  1. #1
    ero
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    interior points

    hello! im having some trouble with this question which is due in tomorrow:

    is it necessarily true that
    int(AuB)=int(A) u int(B)?

    im not sure how to construct a proof as i seem to keep just stating it with out proving it. Please help!
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  2. #2
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    Let A = \{ 0\leq x \leq 1 | x \in \mathbb{Q}\} and B = \{ 0\leq x \leq 1 |x\not \in \mathbb{Q} \}. Then A \cup B = [0,1] and so \mbox{int}(A\cup B) = (0,1) but \mbox{int}(A) = \{ \} and \mbox{int}(B) = \{ \}.
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  3. #3
    ero
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    how is int(AuB)= (0,1) surely int(AuB) would equal {} as well . i think i need to prove that this statement is true?
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  4. #4
    ero
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    sorry i forgot to mention that A and B are subsets of R.(real numbers)
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  5. #5
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    You have been shown that the proposition is false.
    Here is a more mundane example.
    A = [0,1)\,\& \,B = [1,2)\quad  \Rightarrow \quad A \cup B = [0,2)
    Int(A) = (0,1)\,,\,Int(B) = (1,2)\,\& \,Int(A \cup B) = (0,2)

    BUT (0,1) \cup (1,2) \ne (0,2)
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