non compact nested closed sets in metric spaces.

Question

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?

Answer 1

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

Answer 2

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.