Is there a continuous bijection from a segment of $\mathbb{R}$ to the following subset of $\mathbb{Q}_2$?

Question

Is there a continuous bijection from a segment of $\mathbb{R}$ to the following subset of $\mathbb{Q}_2$?

For any element $x$ of $\mathbb{Q}_2$ let $\displaystyle x=2^z\frac{p}{q}r$ where $z\in\mathbb{Z}, p, q$ coprime odd, and $r$ contains the residual factors.

If I define a subset of $\mathbb{Q}_2$ as follows:

$X=\{x\in\mathbb{Q}_2:q=1\}$

Does there exist a continuous bijection from $\mathbb{R}$ or from a segment of $\mathbb{R}$ to $X$?

I may need to define "continuous". If so, I need a bijection such that it maps a continuous function in $\mathbb{R}$ to a continuous function in $(X,\lvert\cdot\rvert_2)$. Hopefully this does that.

Answer 1

$\mathbb{Q}_2$ is totally disconnected, so there is no nonconstant continuous map from any interval in $\mathbb{R}$ to $\mathbb{Q}_2$.

(Incidentally, your belief is not correct, and the set $X$ you refer to cannot be defined in any natural way unless you mean for it to be all of $\mathbb{Q}_2$. Any element of $\mathbb{Q}_2\setminus\mathbb{Q}$ can be written in that form in more than one way: if one value of $r$ works, so does $sr$ for any nonzero rational $s$ by modifying $z$, $p$, and $q$ to compensate.)