What part of the 2-adic space is, and isn't, projected down to the odd numbers by the equivalence relation $x\sim \frac{x}{2}$?
If we define $\mathbb{N}/\sim$ as the partitioning of $\{x\in\mathbb{N_{>0}}\}$ by the equivalence relation $x\sim\frac{x}{2}$, then this effectively projects $\mathbb{N}$ down to the odd numbers.
In fact going further it also projects every fraction $\left\{\frac{2n+1}{2^m}:m,n\in\mathbb{Z}\right\}$ down to an odd number.
All these numbers; am I right in thinking these are best described as $\mathbb{Z}_2\cap\mathbb{Q}$ ?
What else do we find in $\mathbb{Z}_2$ and $\mathbb{Q}_2$?
I'm not clear on what for example would be the value of $\lvert\sqrt{2}\rvert_2$; whether we can have non-integer values such as $\lvert\sqrt{2}\rvert_2=\frac{1}{2}$ which would not be in the above set, but would be in:
$\left\{\frac{2n+1}{2^m}:n\in\mathbb{Z},m\in\mathbb{Q}\right\}$
Any $2$-adic number $x \in \mathbb{Q}_2$ can be uniquely written as
$$x = 2^n u$$
Where $n \in \mathbb{Z}$ is the $2$-adic valuation of $x$ and $u \in 1+2\mathbb{Z}_2$ is an odd $2$-adic integer.
If $x \in \mathbb{Q}$ is a rational number then $u$ is rational, which means that we can write $u=\frac{a}{b}$ with $a$ and $b$ odd.
As for what "happens" to $\sqrt{2}$, or more precisely what are the equivalence classes in $\mathbb{Q}[\sqrt{2}]$, you can understand it by looking at $\mathbb{Q}_2[\sqrt{2}]$. Here, using Hensel's lemma, any element $x \in \mathbb{Q}_2[\sqrt{2}]$ can be uniquely written
$$x = \sqrt{2}^n u$$
with $n \in \mathbb{Z}$ and $u \in 1+ \sqrt{2}\mathbb{Z}_2[\sqrt{2}]$. And when $x\in \mathbb{Q}[\sqrt{2}]$, then $u$ can be written $u = a + b\sqrt{2}$ with $a$ a rational numbers with odd numerator and denominator.