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 $\bigcap {\cal C}\in\tau$?



Connected $T_2$-space such that the open sets are closed under countable intersection
In other words, you want every $G\_\delta$ to be open. I just learned that such spaces are called [**P-spaces**](https://www.encyclopediaofmath.org/index.php?title=P-space). 


The following paper contains an example of a connected Hausdorff P-space (and yes it is uncountable).



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> *Misra, Arvind K.*, [**A topological view of P-spaces**](http://dx.doi.org/10.1016/0016-660X(72)90026-8), General Topology Appl. 2, 349-362 (1972). [ZBL0249.54019](https://zbmath.org/?q=an:0249.54019). [MR317304](https://mathscinet.ams.org/mathscinet-getitem?mr=317304).
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The author also mentions that you can't improve much upon "Hausdorff", because every [functionally Hausdorff](https://topospaces.subwiki.org/wiki/Functionally_Hausdorff_space) P-space is totally disconnected. 


