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

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In other words, you want every $G_\delta$ to be open. I just learned that such spaces are called P-spaces.

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

Misra, Arvind K., A topological view of P-spaces, General Topology Appl. 2, 349-362 (1972). ZBL0249.54019. MR317304.

The author also mentions that you can't improve much upon "Hausdorff", because every functionally Hausdorff P-space is totally disconnected.

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    $\begingroup$ Note that if $X$ is a P-space and $f:X \to \mathbb{R}$ is continuous then $f$ is constant on the connected components of $X$. From this it follows that functionally Hausdorff P-spaces are totally disconnected. Also note that any separable $T_1$ P-space is countable and discrete; in particular it follows that a connected $T_1$ P-space must have uncountable density character. $\endgroup$ – Ramiro de la Vega May 18 '18 at 12:07

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