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 :)

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 , let .

2) If , let .

Re: Show that the unit sphere is closed

Quote:

Originally Posted by

**sakuraxkisu** 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 the complement.

If then find the distance. from to .

That can be done using analytic geometry. Now let .

The ball

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.

Re: Show that the unit sphere is closed

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.

Re: Show that the unit sphere is closed

Quote:

Originally Posted by

**sakuraxkisu** 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.

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?

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 ".

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: .

Re: Show that the unit sphere is closed

Oh, and I should add, another way to describe the points of the sphere:

Re: Show that the unit sphere is closed

Ofcourse, you could also define the (continuous) map and use the fact that

.

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).

Re: Show that the unit sphere is closed

Thank you everyone for all your help! :)