This question has a trivial starting point: If the open sets of $\mathbb{R}$ were closed under countable intersection, the Euclidean topology would be discrete because for all $x\in\mathbb{R}$ we have $\{x\} = \bigcap\big\{ (x-\frac{1}{n},x+\frac{1}{n}):n\in\mathbb{N}\setminus\{0\}\big\}$.

**Question.** Is there a connected $T_2$-space $(X,\tau)$ with $X$ uncountable such that for all countable sets ${\cal C} \subseteq \tau$ we have $\bigcap {\cal C}\in\tau$?