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Math Help - Open Set

  1. #1
    Super Member Showcase_22's Avatar
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    Open Set

    Show that U=\{ (x,y,z) \in \mathbb{R}^3 \ : \ \frac{e^{x+y^2-z}-1}{x^2+y^2-z^3}>7 \} is open in \mathbb{R}^3.
    U can be rearranged to get:

    U=\{(x,y,z) \in \mathbb{R}^3 \ : \ e^{x^2+y^2-z}-1-7x^2-7y^2+7z^3>0 \}

    The definition of an open ball is (\forall u \in U)(\exists \varepsilon_w>0)[B_{\varepsilon_w}(u,d) \subset U] where w \in U.

    Define \varepsilon_w:=\frac{1}{2} \left(e^{x+y^2-z}-1-7x^2-7y^2+7z^3 \right)

    The fact that it is halved means that any point in any ball around a point u will be greater than 0 and will therefore be in U. This ball is open.

    Since this is valid \forall u \in U, U is expressed as a union of open balls ie. \cup_{w \in U} B_{\varepsilon_w}(u,d)=U

    Hence U is open.

    Is this right? I'm wondering if I have made any syntax errors!
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  2. #2
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by Showcase_22 View Post
    U can be rearranged to get:

    U=\{(x,y,z) \in \mathbb{R}^3 \ : \ e^{x^2+y^2-z}-1-7x^2-7y^2+7z^3>0 \}

    The definition of an open ball is (\forall u \in U)(\exists \varepsilon_w>0)[B_{\varepsilon_w}(u,d) \subset U] where w \in U.

    Define \varepsilon_w:=\frac{1}{2} \left(e^{x+y^2-z}-1-7x^2-7y^2+7z^3 \right)

    The fact that it is halved means that any point in any ball around a point u will be greater than 0 and will therefore be in U. This ball is open.

    Since this is valid \forall u \in U, U is expressed as a union of open balls ie. \cup_{w \in U} B_{\varepsilon_w}(u,d)=U

    Hence U is open.

    Is this right? I'm wondering if I have made any syntax errors!
    In a metric space \left(X,d\right) as et O is open if and only if for every x\in O there exists a \delta>0 such that B_{\delta}(x)\subseteq O. Now, have you done that?

    I am honestly not even sure what you are doing here. Are we working with the usual metric?

    For your definition of an open ball, what is \varepsilon_w, I don't understand the w part?
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  3. #3
    Super Member Showcase_22's Avatar
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    I'm trying to show that my set is a union of open balls so it's open. I'm not sure if it's the easiest way of doing it, but I think it might work.

    I'm afraid that my question does not say. Is it possible to prove it for every metric? I not, it probably means Euclidean.

    I was thinking that for every point (x,y,z) e^{x+y^2-z}-1-7x^2-7y^2+7z^3<br />
will produce a single number. If I create a ball of radius <br /> <br />
\delta=\frac{1}{2} \left(e^{x+y^2-z}-1-7x^2-7y^2+7z^3 \right)<br />
and centre (x,y,z), (the B_{\delta}(x,y,z) that you wrote) then this will be an open ball that is always a subset of U.

    Is there something wrong with my idea?
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  4. #4
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by Showcase_22 View Post
    I'm trying to show that my set is a union of open balls so it's open. I'm not sure if it's the easiest way of doing it, but I think it might work.

    I'm afraid that my question does not say. Is it possible to prove it for every metric? I not, it probably means Euclidean.

    I was thinking that for every point (x,y,z) e^{x+y^2-z}-1-7x^2-7y^2+7z^3<br />
will produce a single number. If I create a ball of radius <br /> <br />
\delta=\frac{1}{2} \left(e^{x+y^2-z}-1-7x^2-7y^2+7z^3 \right)<br />
and centre (x,y,z), (the B_{\delta}(x,y,z) that you wrote) then this will be an open ball that is always a subset of U.

    Is there something wrong with my idea?
    What you are saying is analogous to this. Let O=\left\{x\in\mathbb{R}:x^3>0\right\}. To see that this is open we merely note that choosing \frac{x^3}{2}=\delta guarantees that B_{\delta}(x)\subseteq O. Take x=2\implies \delta=4. Then, B_\delta\left(2\right)\nsubseteq O
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  5. #5
    Super Member Showcase_22's Avatar
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    ah, so I was wrong!

    In your example of <br /> <br />
O=\left\{x\in\mathbb{R}:x^3>0\right\}<br />
, does \delta=\frac{x}{2} work in the ball B_{\delta}(x)?

    It works in the case x=2. The set is bounded below by 0 so if a \in O, then \frac{a}{2} is always in the set since it's the same distance from a as it is 0.

    So extending this idea to the set U gives \delta= \frac{\frac{e^{x+y^2-z}-1}{x^2+y^2-z^3}-7}{2}.

    I think i'm starting to get it!
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