# Sequence of closed sets

1. Nov 27, 2008

### hypermonkey2

I was reading about the Nested sphere theorem and a thought occurred. if you have a sequence of decreasing closed sets whose diameter goes to zero in the limit,
we can show that the intersection of all these sets is a single point.

my idea was to show this using nested sphere theorem if we can say that for any closed set we can find a smallest closed sphere containing this set as well as a biggest sphere contained in this set.
that way we can trap our original sequence between two sequences of decreasing spheres.

Is this if fact possible? if so/not, why?

cheers!

2. Nov 28, 2008

### hypermonkey2

essentially, my question can be boiled down to:

for any closed set, can we find a smallest closed sphere containing it? what about a smallest closed sphere contained IN it?

3. Nov 28, 2008

### Office_Shredder

Staff Emeritus
Consider instead of something like the interval [0,1/n] we look at the union of ALL intervals of the form

$$\bigcup [k,k+1/n] = K_n$$ (over R of course)

then the limit of the intersection of these $$\bigcap K_n$$... this is Z which has measure 0, even though each Kn has unbounded measure.

EDIT: Obviously we could make this conform to limn->infinitym(Kn) = 0 by using just $$[1,1+1/n] \cup [0,1/n]$$

4. Nov 28, 2008

### hypermonkey2

thanks for the reply! it is true what you write, but i am having trouble making the connections...

What does this imply?

cheers!

5. Nov 28, 2008

### Office_Shredder

Staff Emeritus
You said
This isn't true. It's if you have a sequence of decreasing closed spheres that this happens... so when trying to extend the result, you get examples like I gave. The first one in fact isn't contained in any sphere. You'll need a boundedness condition to get anywhere

6. Nov 28, 2008

### hypermonkey2

of course! yes i apologize i mean in a complete metric space where the diameter of each set in the sequence is bounded.

Thanks for pointing that out :D

so how would i go about finding these spheres?

7. Nov 30, 2008

### hypermonkey2

is it even reasonable to say that any closed set in this space will have a closed sphere containing it?
And if so, can we simply define a set this way and take the infimum along the radii?

8. Dec 2, 2008