Where can I learn more about the topology on $\mathbb{R}$ induced by the map $\mathbb{R} \to \prod_{a>0} (\mathbb{R}/a\mathbb{Z})$?

Question

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, 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.

Answer 1

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. Berlin-Göttingen-Heidelberg: Springer-Verlag. VIII, 519 p. (1963). ZBL0115.10603.

[2] Comfort, W. Wistar; Hernández, Salvador; Trigos-Arrieta, F. Javier, Relating a locally compact Abelian group to its Bohr compactification, Adv. Math. 120, No. 2, 322-344 (1996). ZBL0863.22004.

[3] Hernández, Salvador, The dimension of an LCA group in its Bohr topology, Topology Appl. 86, No. 1, 63-67 (1998). ZBL0935.22006.

[4] Hernández, Salvador; Remus, Dieter; Javier Trigos-Arrieta, F., Contributions to the Bohr topology by W.W. Comfort, Topology Appl. 259, 28-39 (2019). ZBL1414.22012.

[5] See MR 40:4685.

[6] Glicksberg, I., Uniform boundedness for groups, Can. J. Math. 14, 269-276 (1962). ZBL0109.02001.

[7] Zlatoš, Pavol, The Bohr compactification of an abelian group as a quotient of its Stone-Čech compactification, Semigroup Forum 101, No. 2, 497-506 (2020). ZBL1471.22001.

[8] Zbl 0795.22003, Zbl 1355.37048