I'm studying the dual group of the diadic rationals, $\widehat{\mathbb{Z}[1/2]}$, where the dual is the dual of Pontryagin of $\mathbb{Z}[1/2]$. In some papers says that $\widehat{\mathbb{Z}[1/2]}$, is naturally a compact metric space. The problem is, that I don't know which metric I have to use for this.
Can you help me?
Greetings !
You have an example of a solenoid, the inverse limit of a system of continuous endomorphisms of the unit sphere $S^1$. In particular, you are looking at $\varprojlim S^1$ under the squaring map. Concretely, this is because any continuous map $f:\Bbb Z[\frac{1}{2}]\to S^1$ is determined by the values $f$ takes on powers of two, in particular the sequence $f(\frac{1}{2}),f(\frac{1}{4}),f(\frac{1}{8}),\cdots$ in which each term is a square of the next term, and addition/inversion of $f$ correspond to pointwise operations on sequences.
More abstractly, the dual of any direct limit ($\Bbb Z[\frac{1}{2}]=\varinjlim\frac{1}{2^k}\Bbb Z$) is a corresponding inverse limit; in our case the division-by-two maps become multiplication by two, and the dual of $\Bbb Z$ is $S^1$.
Solenoids have a nice geometric construction: here, take a torus in $\Bbb R^3$, within which is another torus that wraps around the original twice, and within that one is one that wraps around twice, and so on. Take the intersection of all of these tori to get the solenoid. Wikipedia:
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Each individual stage describes a closed set (say we include the boundaries of the tori), and any intersection of closed sets is also closed. Thus, our solenoid is closed and bounded in $\Bbb R^3$, so by invoking the Heine-Borel theorem we conclude it is compact. Indeed, it inherits its topology and metric from the ambient three-dimensional Euclidean space.
(For a justification as to why the inverse limit is homeomorphic to this intersection, I have to defer to experienced topologists. I'd be interested to hear about any generalized or pretty arguments.)