Collections of Bounded, Closed Subsets

• Nov 8th 2010, 04:47 PM
matt.qmar
Collections of Bounded, Closed Subsets
Hello,

There is a theorem that states,

If there is a collection of compact subsets $\displaystyle \{K_i\}$ of a metric space $\displaystyle X$ which statifies the Finite Intersection Property, then $\displaystyle \cap_i\{K_i\} \neq \emptyset$.

I am trying to find examples of collections of subsets from the reals,
where the words
1) closed
2) bounded
replace the word "compact" in the above statement, and show that the theorem is not true for either closed or bounded subsets, so the theorem only holds for compact subsets.

That would imply, there exists collections of (1) closed or (2) bounded subsets whose intersection IS the null set.

Thanks! Hope the question is clear... I am seeking two counterexamples, one each for collections of closed subsets and collections of bounded subsets.

Cheers!
• Nov 8th 2010, 05:07 PM
roninpro
For (1), how about the collection $\displaystyle [1,3], [2,4], \mathbb{R}$?
• Nov 8th 2010, 05:08 PM
TheEmptySet
If I understand correctly this should do for both

$\displaystyle S_n=[2n-1,2n]$ for $\displaystyle n \in \mathbb{Z}$
• Nov 8th 2010, 08:03 PM
matt.qmar
I see that $\displaystyle S_n=[2n-1,2n]$ is closed,
however, I am not sure that it is bounded?

Thank you for the replies!
• Nov 8th 2010, 08:04 PM
matt.qmar
Sorry! Of course it is bounded, for a finite number of n. Yikes. Thank you!
• Nov 8th 2010, 10:08 PM
Drexel28
Quote:

Originally Posted by matt.qmar
Hello,

There is a theorem that states,

If there is a collection of compact subsets $\displaystyle \{K_i\}$ of a metric space $\displaystyle X$ which statifies the Finite Intersection Property, then $\displaystyle \cap_i\{K_i\} \neq \emptyset$.

I am trying to find examples of collections of subsets from the reals,
where the words
1) closed
2) bounded
replace the word "compact" in the above statement, and show that the theorem is not true for either closed or bounded subsets, so the theorem only holds for compact subsets.

That would imply, there exists collections of (1) closed or (2) bounded subsets whose intersection IS the null set.

Thanks! Hope the question is clear... I am seeking two counterexamples, one each for collections of closed subsets and collections of bounded subsets.

Cheers!

I personally like to consider $\displaystyle \mathbb{R}-B_{n}}(0)$. These are all closed and unbounded and their intersection is null.