Connected and Hausdorff topological space whose topology is stable under countable intersection,

There is an example in Arvind K. Misra, A topological view of P-spaces, General Topology and its Applications, Volume 2, Issue 4, December 1972, 349-362. It starts with the space $E_0$ that he constructs in Example $bf{3.1}$, a Hausdorff $P$-space (i.e., one in which $G_delta$-sets are open) with two points $a$ and $b$ that cannot be separated by a continuous function. In Example $bf{5.3}$ he recursively constructs from $E_0$ spaces $E_n$ for $ninomega$ in such a way that $E_n$ is embedded in $E_{n+1}$ and then defines $E_omega$ to be the direct limit of the sequence $langle E_n:ninomegarangle$. (The topology on $E_omega$ is the final topology determined by the embeddings.) $E_omega$ is a Hausdorff $P$-space on which every real-valued continuous function is constant, so it is connected.

In any $P$-space every convergent sequence is eventually constant.