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Math Help - Set A open, but f(A) not open

  1. #1
    Member thaopanda's Avatar
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    Set A open, but f(A) not open

    Let f : R^2 \rightarrow R given by f(x,y) := x^2. Find a set A open in R^2 such that f(A) in not open in R.

    I don't get how to do this...
    help..?
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  2. #2
    Moo
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    Hello,

    Is it really x^2 ?

    If so, you can just take A=(-a,a) \times \mathcal{O}, where \mathcal{O} is any open set in \mathbb{R} and a>0

    Then f(A)=[0,a^2) which is not an open set.
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  3. #3
    Member thaopanda's Avatar
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    is that really all there is to it? I just assume all my answers are supposed to be super long, 'cause so far, most of them have been taken up half a page... hehe now I feel silly

    Thank you very much!
    Nicole
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    MHF Contributor arbolis's Avatar
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    Quote Originally Posted by Moo View Post
    Hello,

    Is it really x^2 ?

    If so, you can just take A=(-a,a) \times \mathcal{O}, where \mathcal{O} is any open set in \mathbb{R} and a>0

    Then f(A)=[0,a^2) which is not an open set.
    Hey Moo I'm curious. Why did you chose A=(-a,a) \times \mathcal{O} and not A=(-a,a)? I don't really know what your " \times" product means. I understand it as "multiplying" two sets to get another one... but I don't know if it makes sense.
    Wouldn't A=(-a,a) be sufficient?
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  5. #5
    Moo
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    Quote Originally Posted by thaopanda View Post
    is that really all there is to it? I just assume all my answers are supposed to be super long, 'cause so far, most of them have been taken up half a page... hehe now I feel silly

    Thank you very much!
    Nicole
    I guess that's all. Unless I made a mistake in the definition of an open set in \mathbb{R}^2 ^^'

    Quote Originally Posted by arbolis View Post
    Hey Moo I'm curious. Why did you chose A=(-a,a) \times \mathcal{O} and not A=(-a,a)? I don't really know what your " \times" product means. I understand it as "multiplying" two sets to get another one... but I don't know if it makes sense.
    Wouldn't A=(-a,a) be sufficient?
    Because A has to be an open set in \mathbb{R}^{\color{red}2}.
    \times stands for the cartesian product. Because we're having two values, one in \mathbb{R}, the other one in \mathbb{R}
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  6. #6
    MHF Contributor arbolis's Avatar
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    Quote Originally Posted by Moo View Post

    Because A has to be an open set in \mathbb{R}^{\color{red}2}.
    \times stands for the cartesian product. Because we're having two values, one in \mathbb{R}, the other one in \mathbb{R}
    Thanks, I missed this part!
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