Can $\Bbb Z[\frac16]/{\sim}$ inherit the topology from the 2-adic integers $\Bbb Z_2$, and is it compact?
Where $\sim$ says $x\sim y\iff 2^nx=2^{-n}y$
It seems obvious $\Bbb Z[\frac16]$ doesn't naturally do so, since it contains numbers $x$ satisfying $\lvert x\rvert_2>1$, which aren't in $\Bbb Z_2$
But every class defined by $\Bbb Z[\frac16]/{\sim}$ is either $\{0\}$ or a pair having a unique element in $\Bbb Z[\frac16]>0$ so this claim seems equivalent to asking whether $\Bbb Z[\frac16]\geq0$ can inherit the topology, which it seems to trivially do.
As for whether it's compact, this seems to hang on whether there exists a finite subcover of $\Bbb Z[\frac16]/{\sim}$, and I'm a bit lost in that regard.