# Is there a connected $T_2$-topology on $\mathbb{Q}$ that is coarser than the Euclidean one?

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 $T_2$?

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.