Consider the (continuous, injective, abelian group homomorphism) map $\Phi \colon \mathbb{R} \to \prod\_{a>0} (\mathbb{R}/a\mathbb{Z})$ (where the target is given the product topology) taking $x\in \mathbb{R}$ to the family $(x \mod a) \_ {a>0}$; depending on your inclination, you may prefer to view it as $\Phi \colon \mathbb{R} \to (\mathbb{S}^1)^{\mathbb{R} \_ {>0}}$ taking $x$ to $(\exp(2i\pi b x)) \_ {b>0}$ (with $b=1/a$).


I'd like to know more about the following two topological spaces, indeed topological groups:


* The image of $\Phi$ with the subspace topology, call it $R$, or, equivalently, the coarsest topology on $\mathbb{R}$ making $\Phi$ continuous (this is strictly coarser than the usual topology).
* The closure of the image of $\Phi$ with the subspace topology, call it $S$ (since the target of $\Phi$ is compact, so is $S$).


There seems to be a lot of interesting things to say about $R$ and $S$ in the line of “examples and counterexamples in general topology”; for example, it turns out that [if $(x\_n)$ is a real sequence such that $\Phi(x\_n) \to 0$ then in fact $x\_n \to 0$ in the reals](https://math.stackexchange.com/questions/4590203/if-x-n-to-0-pmoda-for-every-a0-does-it-follow-that-x-n-to-0), so $R$ has the same convergent sequences as $\mathbb{R}$ even though it does not have the same topology (so $R$ is not metrizable). I have a million more questions about $R$, about $S$, and also about the map $\beta\mathbb{R} \to S$ (and maybe about $\beta R \to S$ while I'm at it), but rather than ask a million questions now, let me start by asking what has already been written about them out there in the literature:


**Question:** Do $R$ and $S$ have standard names? Are they described in a paedagogical fashion in some textbook on general topology or such text?


A presentation in the spirit of Steen & Seebach's classic *Counterexamples in Topology* would be a good start (I was unable to find these spaces in the book in question but, of course, I may have missed them). A name would help with the search (this is somewhat related to solenoids, but solenoids are metrizable whereas $R$ and $S$ are not, as I mention above).


Of course, if you have some favorite facts to share about $R$ and $S$, they're welcome as well.



Where can I learn more about the topology on $\mathbb{R}$ induced by the map $\mathbb{R} \to \prod_{a>0} (\mathbb{R}/a\mathbb{Z})$?
Short answer: $S$ is known as the *Bohr compactification* of $\mathbf R$ (often denoted $S=b\mathbf R$, see [1], 26.11), and $R$ is known as $\mathbf R$ with the *Bohr topology* (often denoted $R=\mathbf R^+$, see e.g. [2, 3, 4]).


For a longer answer, write $\chi\_b(x)=\exp(2\pi ibx)$ so that your $\Phi(x)=(\chi\_b(x))\_{b>0}$. Then $R$ is $\mathbf R$ with the weakest topology making $\chi\_b$ continuous for all $b>0$. Since $\chi\_0$ is constant and $\chi\_{-b}$ is $\chi\_b$ composed with complex conjugation, we see that it makes no difference to use $(\chi\_b(x))\_{b\in\mathbf R}$ instead. That shows that your definition coincides with those in *loc. cit.*


More generally if $G$ is any locally compact abelian group with dual $\hat G$ and we write $$\Phi(g)=(\chi(g))\_{\chi\in\hat G},$$ these references define $bG$ as the closure of $\Phi(G)$ in $\mathbf T^{\hat G}$, and $G^+$ as $G$ with its relative topology in $bG$. Favorite facts about $G^+$ are that it is sequentially closed in $bG$ (no sequence in $G^+$ can converge to a point of $bG\smallsetminus G^+$ [5]), and Glicksberg's theorem that $G^+$ has exactly the same compact sets as $G$ [6]. The map $\beta G\to bG$ is also studied in [7].


Finally, *since you ask*, my own favorite facts about the Bohr topology $G^+$ occur for $G=\mathbf R^n$, $n>1$: namely, a parabola is Bohr dense in the plane, and more generally the image of any polynomial map $\mathbf R^m\to \mathbf R^n$ is Bohr dense in its affine hull [8].




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[1] *Hewitt, E.; Ross, K. A.*, [**Abstract harmonic analysis. Vol. I.**](//archive.org/details/in.ernet.dli.2015.134308/page/n382/mode/1up) Berlin-Göttingen-Heidelberg: Springer-Verlag. VIII, 519 p. (1963). [ZBL0115.10603](https://zbmath.org/?q=an:0115.10603).


[2] *Comfort, W. Wistar; Hernández, Salvador; Trigos-Arrieta, F. Javier*, [**Relating a locally compact Abelian group to its Bohr compactification**](https://doi.org/10.1006/aima.1996.0042), Adv. Math. 120, No. 2, 322-344 (1996). [ZBL0863.22004](https://zbmath.org/?q=an:0863.22004).


[3] *Hernández, Salvador*, [**The dimension of an LCA group in its Bohr topology**](https://doi.org/10.1016/S0166-8641(97)00126-0), Topology Appl. 86, No. 1, 63-67 (1998). [ZBL0935.22006](https://zbmath.org/?q=an:0935.22006).


[4] *Hernández, Salvador; Remus, Dieter; Javier Trigos-Arrieta, F.*, [**Contributions to the Bohr topology by W.W. Comfort**](https://doi.org/10.1016/j.topol.2019.02.018), Topology Appl. 259, 28-39 (2019). [ZBL1414.22012](https://zbmath.org/?q=an:1414.22012).


[5] See [MR 40:4685](//ams.org/mathscinet-getitem?mr=40:4685).


[6] *Glicksberg, I.*, [**Uniform boundedness for groups**](https://doi.org/10.4153/CJM-1962-017-3), Can. J. Math. 14, 269-276 (1962). [ZBL0109.02001](https://zbmath.org/?q=an:0109.02001).


[7] *Zlatoš, Pavol*, [**The Bohr compactification of an abelian group as a quotient of its Stone-Čech compactification**](https://doi.org/10.1007/s00233-020-10121-6), Semigroup Forum 101, No. 2, 497-506 (2020). [ZBL1471.22001](https://zbmath.org/?q=an:1471.22001).


[8] [Zbl 0795.22003](//zbmath.org/0795.22003), [Zbl 1355.37048](//zbmath.org/1355.37048)


