Manifolds with finitely many ends In the article ' The structure of stable minimal hypersurfaces in $ R^{n+1} $ ( http://arxiv.org/pdf/dg-ga/9709001.pdf) of Cao-Shen-Zhu the remark 2 at page 3 contains a statement that i don't understand (actully it seems me false).
Let $ M $ be a manifold and let $ \{K_n\} $ be an exhaustation by compact sets: 
$$ \cup_n K_n = M $$ and 
$$ K_n \subset K_{n+1} $$
An end of $ M $ is a collection of subsets (actually open subsets) $ \{E_n\} $ such that $ E_n $ is a connected component of $ M-K_n $ and
$$ E_{n+1} \subset E_n $$ 
It can be proved that the number of ends is independent from the choice of the exhaustation $\{K_n\} $
Now in the article above is stated that if a manifold has only finitely many ends $ \{ E_{n}^{1} \}, \ldots \{ E_{n}^{k} \} $ there exists $ n_0 $ such that 
$$ E_{n}^{j} = E_{n_0}^{j} $$
for every $ j = 1 \ldots k $ and $ n \geq n_0 $
This statement seems me false. 
Thanks

You're right, that statement is false. They probably mean that the number of ends stabilizes at some point (as there might be just one component at first, then more, then more, etc.).
