Find in X a sequence of closed sets $(F_n)_{n=1}^\infty$ with the finite intersection property but $\cap_{n=1}^\infty F_n= \emptyset$ The space $X = [0,1]$ with the metric $d(x,y) = |x| + |y|$ for $x \neq y$, $ d(x,x) = 0$ is not compact. Find in X a sequence of closed sets $(F_n)_{n=1}^\infty$ with the finite intersection property, but such that $\cap_{n=1}^\infty F_n= \emptyset$.
Could Cantor set be an example of it?

Note that all sets $\{x\}$ with $x>0$ are open, as the $x$-neighborhood of $x$ only consists of $x.$
Now $F_n=\{1/2+1/k : k> n\}$ does the job. Clearly $\cap_{n=1}^\infty F_n= \emptyset,$ and $F_{i_1}\cap \dots\cap F_{i_p} = F_{\max\{i_1,\dots,i_p\}},$ so the finite intersection property holds.
Also, $F_n$ is closed as $[0,1]\setminus F_n$ is open as the union of the open sets $[0,1/3)$ and $\{x\}$ for $x>1/3.$
