Let $M$ be a connected manifold with precisely $k$ ends $\epsilon_1,...,\epsilon_k$. Choose a collection $(U_i)_{i=1}^k$ of pairwise disjoint open $\epsilon_i$-neighborhoods. Then I wonder how to prove that $N := M \setminus (\bigcup_{i=1}^k U_i)$ is a bounded subset of M, i.e, a set with compact closure.

Thoughts I had so far: It is clear that each component of $N$ is bounded by itself. However, it is quite possible that $N$ consists of infinitley many such components, so an argument on finite union doesnt work. Does anybody have an idea ?

*Edit:* As established in the comments, the problem lies in the fact that there at least two definitions for ends of manifolds that are non-equivalent in general:

I used a version of Siebemanns definition (1965), where an end of $X$ is an equivalence class of sequences $U_1 \supset U_2 \supset U_3...$ of connected open subsets with compact boundary, so that $\bigcap_{i=1}^\infty cl(U_i) = \emptyset$. Two such sequences $(U_i)$ and $(V_i)$ are equivalent if for each $n \in \mathbb N$ there exists $j \in \mathbb N$, so that $U_i \supset V_j$ and $V_i \supset U_j$. The set of corresponding ends is denoted by $\mathcal E_1$.

If $X$ admits a compact exhaustion $(K_i)_{i=1}^\infty$ (for example if $X$ is a manifold), i.e. a sequence of compact subsets $K_1 \subset K_2 \subset ...$ with $\bigcup_{i=1}^\infty int(K_i) = X$, then one can define another set of ends $\mathcal E_2$ as the inverse limit of the system $\pi_0(X\setminus K_1) \leftarrow \pi_0(X \setminus K_2) \leftarrow \pi_0(X \setminus K_2)....$.

Now my original question can be formulated as follows:

*If $X$ is a connected manifold, is there a bijection of sets $f: \mathcal E_1 \to \mathcal E_2$ ?*

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