Let $X=\Bbb Z[\frac16]^+$ be the positive dyadic and ternary rationals, with their topology inherited from $\Bbb R$ and let $p^{\nu_p(x)}$ be the highest power of $p$ that divides $x$.
Now make a topological quotient $X/{\sim}$ of $X\subset \Bbb R$ that says a set $X'\subset X$ is open if for all $x\in X'$ the set also contains $\lim_{n\to\infty}f^n(x)$
Where $f^n(x)=x+(1-2^{-6n})\cdot2^{\nu_2(x)}\cdot3^{\nu_3(x)-1}$
Here $f:\Bbb N\times X\to X$ and $n$ indicates compositions of $f$.
Question
Does $X/{\sim}$ make sense as a quotient space? Is it well-defined? In particular, I'm aware that the condition I've defined is not clearly (to me at least) an equivalence relation and therefore may not define a quotient. Does taking a topology, and then to declare some subsets are open, define some other form of new topology from the original?
Motivation
All being well, under this topology, the Collatz graph will have finitely many connected subgraphs if and only if the quotient space is disconnected. But my topology is limited at best and I need to learn whether all is well.
[A note on graph-connectedness here. Under this topology, graph-connectedness is largely equivalent to topological connectedness.]