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 , let .
2) If , let .
(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
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.
Suppose . Then you would be setting , and the distance from to is . So that ball would have points inside and outside the sphere.
Instead, consider the point as a vector. Let . Let where is the magnitude of . 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.
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.
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 ".
Edit: If you need something beyond that, then yeah, you should break it down into cases:
If and then .
If and then . Subtracting the distance from y to x from both sides we get: .
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).