TITLE: non compact nested closed sets in metric spaces.
QUESTION [6 upvotes]: Do you have an example for closed sets   $...\subseteq F_4\subseteq F_3\subseteq F_2\subseteq F_1$ such that:
$$\bigcap_{n=1}^\infty F_n=\emptyset $$
in $\mathbb{R}^n$ or a metric space?

REPLY [4 votes]: Let $X$ be an unbounded metric space and $x \in X$. Set $F_n=X \setminus B(x,n)$ then $F_n$ is a nested chain of closed sets, which are non-empty by the assumption that $X$ is unbounded. But $\bigcap_{n \in \mathbb N} F_n $ is clearly empty.