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Math Help - Show that the unit sphere is closed

  1. #1
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    Show that the unit sphere is closed

    (note: please ignore the version of this I posted in the pre-university geometry section)

    Hi, my question is:

    Show that the unit sphere with center 0 in R3, namely the set S2 :={x in R3 : (x1)^2+(x2)^2+(x3)^2=1} is closed in R3.

    What I've done so far is say that to prove it's closed, I need to show that the complement of S2, X`S2, is open, so:

    X`S2 :={x in R3 : (x1)^2+(x2)^2+(x3)^2 is not equal to 1}. I then stated the definition of an open set:

    X`S2 is open if for all x in S2 there exists r>0 such that B(x, r) is a subset of S2, B(x, r)={y : d(x, y)<r}.

    What I'm stuck on in this question is the choice of r, and how to show that, for any y in B(x,r), y is in S2.

    Any help would be appreciated
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  2. #2
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    Re: Show that the unit sphere is closed

    Do you understand that if "[tex]x_1^2+ x_2^2+ x_3^2\ne 1[tex]", then either [tex]x_1^2+ x_2^2+ x_3^2> 1[tex] or [tex]x_1^2+ x_2^2+ x_3^2< 1[tex]? So do this in two parts:

    1) If x_1^2+ x_2^2+ x_3^2> 1, let d= x_1^2+ x_2^2+ x_3^2- 1.

    2) If x_1^2+ x_2^2+ x_3^2< 1, let d= 1- x_1^2+ x_2^2+ x_3^2.
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  3. #3
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    Re: Show that the unit sphere is closed

    Quote Originally Posted by sakuraxkisu View Post
    Show that the unit sphere with center 0 in R3, namely the set S2 :={x in R3 : (x1)^2+(x2)^2+(x3)^2=1} is closed in R3.

    What I've done so far is say that to prove it's closed, I need to show that the complement of S2 is open, so:
    X`S2 :={x in R3 : (x1)^2+(x2)^2+(x3)^2 is not equal to 1}. I then stated the definition of an open set:
    X`S2 is open if for all x in S2 there exists r>0 such that B(x, r) is a subset of S2, B(x, r)={y : d(x, y)<r}.
    What I'm stuck on in this question is the choice of r, and how to show that, for any y in B(x,r), y is in S2.
    For notation say that C=(S^2)^c the complement.

    If P: (a,b,c)\in C then find the distance. \delta>0 from P to S^2.
    That can be done using analytic geometry. Now let r=\tfrac{\delta}{2}.
    The ball \frak{B}(P;r)\cap S^2=\emptyset.
    Thanks from sakuraxkisu
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  4. #4
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    Re: Show that the unit sphere is closed

    Yeah, having two different cases makes sense. But I was wondering, for case 2), did you mean d=1-((x1)^2+(x2)^2+(x3)^2) or d=1-(x1)^2+(x2)^2+(x3)^2 ? Thank you.
    Last edited by sakuraxkisu; October 23rd 2013 at 01:35 PM.
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  5. #5
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    Re: Show that the unit sphere is closed

    Quote Originally Posted by HallsofIvy View Post
    Do you understand that if "[tex]x_1^2+ x_2^2+ x_3^2\ne 1[tex]", then either [tex]x_1^2+ x_2^2+ x_3^2> 1[tex] or [tex]x_1^2+ x_2^2+ x_3^2< 1[tex]? So do this in two parts:

    1) If x_1^2+ x_2^2+ x_3^2> 1, let d= x_1^2+ x_2^2+ x_3^2- 1.

    2) If x_1^2+ x_2^2+ x_3^2< 1, let d= 1- x_1^2+ x_2^2+ x_3^2.
    Suppose x_1=x_2=2, x_3=1. Then you would be setting d=4, and the distance from (2,2,1) to (0,0,0) is \sqrt{5}<4. So that ball would have points inside and outside the sphere.

    Instead, consider the point (x_1,x_2,x_3) as a vector. Let \vec{x} = (x_1,x_2,x_3). Let r = \left|\dfrac{|\vec{x}|-1}{2}\right| where |\vec{x}| is the magnitude of \vec{x}. That radius will guarantee you remain in the respective "inside" or "outside" of the sphere.

    Edit: This is actually the method that Plato just posted.
    Last edited by SlipEternal; October 23rd 2013 at 01:17 PM.
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  6. #6
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    Re: Show that the unit sphere is closed

    Thank you for your help. I'm having a bit of trouble showing that, for case 1, where (x1)^2+(x2)^2+(x3)^2 >1, any y in B(x, r) is in S2. I know that I need to get:
    y=sqrt((y1)^2+(y2)^2+(y3)^2) > 1
    I'm not sure how to though, would you be able to help me? Thank you.
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  7. #7
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    Re: Show that the unit sphere is closed

    Quote Originally Posted by sakuraxkisu View Post
    Thank you for your help. I'm having a bit of trouble showing that, for case 1, where (x1)^2+(x2)^2+(x3)^2 >1, any y in B(x, r) is in S2. I know that I need to get:
    y=sqrt((y1)^2+(y2)^2+(y3)^2) > 1
    I'm not sure how to though, would you be able to help me? Thank you.
    If you follow Plato's advice and take the minimum distance from your point to a on the sphere and divide it by 2, there is no need to break down cases. The formula I gave you does just that.
    Last edited by SlipEternal; October 23rd 2013 at 02:38 PM.
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  8. #8
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    Re: Show that the unit sphere is closed

    So then how would you show that, for any y in B(x, r), y is in S2, using the formula for r that you gave me instead of the 2 cases?
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  9. #9
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    Re: Show that the unit sphere is closed

    I would say, "Because we chose a radius that is smaller than the minimum distance between the point and the sphere, all points in our ball are in \mathbb{R}^3\setminus S^2".

    Edit: If you need something beyond that, then yeah, you should break it down into cases:

    If \vec{x}\in\mathbb{R}^3\setminus S^2 and |\vec{x}|<1 then d(\vec{0},\vec{y}) \le d(\vec{0},\vec{x}) + d(\vec{x},\vec{y}) < |\vec{x}| + \left|\dfrac{1-|\vec{x}|}{2}\right| = \dfrac{1+|\vec{x}|}{2} < 1.

    If \vec{x}\in\mathbb{R}^3\setminus S^2 and |\vec{x}|>1 then d(\vec{0},\vec{x})\le d(\vec{0},\vec{y}) + d(\vec{y},\vec{x}). Subtracting the distance from y to x from both sides we get: d(\vec{0},\vec{y}) \ge d(\vec{0},\vec{x}) - d(\vec{y},\vec{x}) = |\vec{x}| - \left|\dfrac{1-|\vec{x}|}{2}\right| = |\vec{x}| + \dfrac{1-|\vec{x}|}{2} = \dfrac{1+|\vec{x}|}{2} > 1.
    Last edited by SlipEternal; October 23rd 2013 at 03:03 PM.
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  10. #10
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    Re: Show that the unit sphere is closed

    Oh, and I should add, another way to describe the points of the sphere: S^2 = \{\vec{x}\in\mathbb{R}^3: |\vec{x}| = 1\}
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  11. #11
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    Re: Show that the unit sphere is closed

    Ofcourse, you could also define the (continuous) map f(x_1,x_2,x_3)=x_1^2+x_2^2+x_3^2-1 and use the fact that
    S^2=f^{-1}(\{0\}).
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  12. #12
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    Re: Show that the unit sphere is closed

    you could also prove it proving that is a compact, in other words that for every open cover you could find a finite open subcover. If you show this (is not difficoult) you can prove that is closed (compacts in T2 spaces are closed).
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  13. #13
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    Re: Show that the unit sphere is closed

    Thank you everyone for all your help!
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