Examples of decreasing sequences of closed sets with constant diameter and empty intersection in complete metric spaces

Question

Looking through older exams from the topology class I'm taking, I found an interesting problem.

Give an example: $ (X, d) $ - a complete metric space

$ F_1 \subset F_2 \subset F_3 \subset ... $ - a decreasing sequence of closed sets with $ diam(F_n) = 1 $ for all $ n $

$ \bigcap F_n = \emptyset $

I found an easy example to this, but it's also quite boring: We can take natural numbers with the discrete metric and $ F_n = \mathbb{N} - \{1, ..., n-1\} $.

Are there any more interesting examples, particularly not involving the discrete metric?

Answer 1

Take $F_n=[n,\infty)$ in the real line with the metric $d(x,y)=\min(|x-y|,1)$.

Note that this metric gives the standard topology on the line.

More generally, take any unbounded complete metric space $(X,d)$, fix a point $x_0\in X$, let $F_n=\{x:d(x,x_0)\ge n\}$. These are unbounded closed sets with empty intersection. Now truncate the metric by $1$ as above, keeping the same sets $F_n$. Their diameters become equal to $1$.