# non compact nested closed sets in metric spaces.

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?

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Try $F_n = [n,\infty)$ in $\mathbb{R}$.
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.