# descending sequence proof

• Oct 20th 2009, 02:05 PM
ElieWiesel
descending sequence proof
Hi, I need this problem to study off of. It is not for a grade. Please provide a complete prove if you can. Thanks,

Suppose $K_1 \supset K_2 \supset K_3 \supset$... is a descending sequence of nonempty sonnected subsets of a metric space M. Prove: if M is compact, the intersection $\bigcap K_n$ is a connected subset of M.

Also give a counterexample to show this is false without compactness.

Thanks.
• Oct 21st 2009, 12:37 AM
Opalg
Quote:

Originally Posted by ElieWiesel
Hi, I need this problem to study off of. It is not for a grade. Please provide a complete prove if you can. Thanks,

Suppose $K_1 \supset K_2 \supset K_3 \supset$... is a descending sequence of nonempty sonnected subsets of a metric space M. Prove: if M is compact, the intersection $\bigcap K_n$ is a connected subset of M.

See this thread for some hints on how to do a similar problem.

Quote:

Originally Posted by ElieWiesel
Also give a counterexample to show this is false without compactness.

In the space $\mathbb{R}^2$ let $A_n = \{(x,y):|x|<1,\ y, and let $K_n$ be the complement of $A_n$, $K_n = \mathbb{R}^2\setminus A_n$. Then $K_n$ is the whole of $\mathbb{R}^2$ except for a strip of width 2 around the y-axis going from $-\infty$ up to the height y=n. This gives a descending sequence of nonempty connected subsets $K_n$ whose intersection is the whole of $\mathbb{R}^2$ except for the entire strip of width 2 around the y-axis. That is clearly not a connected set.
• Oct 21st 2009, 09:47 AM
rn443
Edit: Never mind, I gave an incorrect proof.