The subset of the 2-adic units which are also conventional integers $\mathbb{Q}_2^{\times}\cap\mathbb{N}$ is filtered by the function $f(x)=2x+1$, by which I mean if we set the function's domain to $\mathbb{N}_2^{\times}=\{1,3,5,7,9,11,13,\ldots\}$ then only the principal units $U^{(1)}=\{3,7,11,\ldots\}$ are in its range, which is a proper subset of its domain and subsequent applications of $f$ continue to generate further proper subsets of their predecessor.
We can continue indefinitely to generate $U^{(n)}=\{x:x\equiv2^{n+1}-1\mod2^{n+1}\}$ and every odd integer is filtered out at some point.
Does this filter naturally induce a metric/topology on $\mathbb{Q}_2\cap\mathbb{N}$?
The kind of thing I have in mind for my purposes is (but I don't want to reinvent the wheel here); define a valuation $x_{2\times}$ such that the valuation of $x$ is the highest $n$ for which $U^{(n)}$ contains $x$:
$x_{2\times}=\max\{n:x\in U^{(n)}\}$
Then the distance metric $d(x,y)$ says two odd numbers are close if the $n$ at which they leave the order of units is close:
$d(x,y)=\lvert x_{2\times}-y_{2\times}\rvert$
My motivation to some degree is to define a metric which is orthogonal to the 2-adic metric. I guess I'm asking if this makes sense, does it define a meaningful topology, and is this a rehashing of something which already exists which I may be able to study rather than start from scratch?
Also, I use the example of 2-adics but I will need to use this in the 2, 3 and 4-adic metric spaces so I'm not just looking for the obvious modular arithmetic answer for the special case of the 2 adics.
EDIT: I now realise it defines a pseudometric. Does this form a basis for some topology?