1
$\begingroup$

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?

$\endgroup$
2
$\begingroup$

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 $x>1/3.$

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.