Find a topological space $(X, \tau_X)$ and a sequence of non-empty nested compact sets such that their intersection is empty. An example that was given to me was to take $X=\Bbb N$ with the co-finite topology and $C_n=[n,\infty)$ and we in fact have $C_{n+1} \subseteq C_{n}$ and $\bigcap_{n \in \Bbb N} C_n = \emptyset$.
Since this was just given to me, I made no effort to find such an example. I want to find one by myself but I am out of ideas. Can someone give a hint or bring some intuition ?
(I also know that such sets must not be closed because of Cantor's theorem).
Take $X=\mathbb{R}$ with trivial topology. And $A_n \subset X$ any sequence of sets such that $\bigcap_n A_n =\emptyset $ and such that $A_{n+1} \subset A_{n} .$
For example $A_n =[ n, \infty ) $ or $A_n =\left(0,\frac{1}{n}\right)$ etc,