https://math.hws.edu/eck/metric-spaces/open-and-closed-sets.html
Please refer to Exercise 1.4 on the above link.
Consider the metric space $(\mathbb{R}, d)$. For $n=$1,2,3… let $\mathcal{O}_n$
be the open set $(1, 1+1/n)$
Show that {$\mathcal{O}_n | n=1,2,3…$}
is an infinite collection of open sets whose intersection is not open.
I think that the intersection of the above sets should be empty set, which is open set. So why is the intersection not open?
Best Answer
You are right (twice): the intersection is empty and the empty set is an open set.
The statement would be correct if $\mathcal O_n=\left(1-\frac1n,1+\frac1n\right)$, since then $\bigcap_{n\in\Bbb N}\mathcal O_n=\{1\}$.