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 $T_2$?

$\begingroup$ Possible duplicate of Countable connected Hausdorff space $\endgroup$ – Alex M. May 19 '18 at 15:16

$\begingroup$ Several more examples are to be found in a related MSE post. $\endgroup$ – Alex M. May 19 '18 at 15:17
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 wellknown (and can be easily shown using the Chinese Remainder Theorem) that the Golomb topology is Hausdorff and connected.

$\begingroup$ Why do you call this "Furstenberg topology"? it seems it's plainly the profinite topology. $\endgroup$ – YCor May 19 '18 at 15:04

$\begingroup$ Why do you answer the question instead of closing it as a duplicate of the other question where you yourself have given the accepted answer? $\endgroup$ – Alex M. May 19 '18 at 15:18

$\begingroup$ @AlexM. Yes, you are right. Simply I do not remember all answers I give though I realize that this is very close to what was have been discussed earlier (especially the Golomb topology). I thought that the system shows all the related questions and answers at the moment of posing the question. So, the author always has a possibility to look and analyse what has already been asked and answered prior to posting his own question. $\endgroup$ – Taras Banakh May 19 '18 at 15:22

1$\begingroup$ @YCor The term "Furstenberg topology" is standard and wellaccepted in this area, see the seminal paper of Golomb (dml.cz/bitstream/handle/10338.dmlcz/700933/…) and the recent paper (alpha.math.uga.edu/~pete/CLLP_November_30_2017.pdf) $\endgroup$ – Taras Banakh May 19 '18 at 15:24

$\begingroup$ @YCor, this is the progroup topology on the nonnegative integers. The full profinite topology is discrete since it allows homomorphisms to any finite monoid not just fine groups. $\endgroup$ – Benjamin Steinberg May 19 '18 at 15:35