Let $\mathbb{Q}$ be the rationals, and let $\tau$ be the Euclidean topology on $\mathbb{Q}$. Is there a topology $\tau' \subseteq \tau$ such that $(\mathbb{Q},\tau')$ is connected and $T\_2$?



Is there a connected $T_2$-topology on $\mathbb{Q}$ that is coarser than the Euclidean one?
Using Sierpinski's topological characterization of $\mathbb Q$ (as a unique countable regular second countable space without isolated points), it can be shown that $\mathbb Q$ is homeomorphic to the set $\mathbb N$ endowed with the *Furstenberg topology* $\tau$ generated by the base consisting of all possible arithmetric sequences $a+\mathbb N\_0b:=\{a+bn:n\ge 0\}$ with $a,b\in\mathbb N$.


The Furstenberg topology contains the *Golomb topology* $\tau'$ on $\mathbb N$, which is generated by the base consisting of arithmetric sequences $a+\mathbb N\_0b$ where $a,b$ are comprime. 


It is well-known (and can be easily shown using the Chinese Remainder Theorem) that the Golomb topology is Hausdorff and connected.


