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 $T_2$?
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asked May 19, 2018 at 13:30
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$\begingroup$ Possible duplicate of Countable connected Hausdorff space $\endgroup$– Alex M.Commented May 19, 2018 at 15:16
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$\begingroup$ Several more examples are to be found in a related MSE post. $\endgroup$– Alex M.Commented May 19, 2018 at 15:17
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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.
answered May 19, 2018 at 14:24
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$\begingroup$ Why do you call this "Furstenberg topology"? it seems it's plainly the profinite topology. $\endgroup$– YCorCommented May 19, 2018 at 15:04
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$\begingroup$ 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.Commented May 19, 2018 at 15:18
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$\begingroup$ @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. $\endgroup$ Commented May 19, 2018 at 15:22
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1$\begingroup$ @YCor The term "Furstenberg topology" is standard and well-accepted 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) $\endgroup$ Commented May 19, 2018 at 15:24
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$\begingroup$ @YCor, this is the pro-group topology on the non-negative integers. The full profinite topology is discrete since it allows homomorphisms to any finite monoid not just fine groups. $\endgroup$ Commented May 19, 2018 at 15:35